Stefan Pokorski-Gauge Field Theories and Errata

Gauge Field Theories,Second Edition

CAMBRIDGE UNIVERSITY PRESS

STEFAN POKORSKI

Gauge Field Theories, Second Edition

Quantum field theory forms the present theoretical framework for our understand-ing of the fundamental interactions of particle physics. This up-dated and expandedtext examines gauge theories and their symmetries with an emphasis on theirphysical and technical aspects.

Beginning with a new chapter giving a systematic introduction to classical fieldtheories and a short discussion of their canonical quantization and the discretesymmetries C , P and T , the book provides a brief exposition of perturbationtheory, the renormalization programme and the use of the renormalization groupequation. It then explores topics of current research interest including chiralsymmetry and its breaking, anomalies, and low energy effective lagrangians andsome basics of supersymmetry. A chapter on the basics of the electroweak theoryis now included.

PROFESSOR POKORSKI, a distinguished theoretical physicist, has presentedhere a self-contained text for graduate courses in physics, the only prerequisiteis some grounding in quantum field theory.

Born in 1942, Professor Stefan Pokorski received his PhD in theoretical physicsin 1967 from Warsaw University, where he is now holder of the Chair in TheoreticalParticle Physics. He has been a member of the Polish Academy of Science since1991 and was Director of the Institute for Theoretical Physics, University ofWarsaw, from 1984 to 1994 and President of the Polish Physical Society from 1992to 1994. A visiting scientist at CERN and the Max-Planck Institute for Physics,Munich, for periods of time totalling almost ten years, Professor Pokorski has alsomade regular visits to universities and laboratories around the world. ProfessorPokorski has had many scientific articles published in leading international journalsand in recent years has concentrated on physics beyond the standard model andsupersymmetry.

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Gauge Field TheoriesSecond Edition

STEFAN POKORSKIInstitute for Theoretical Physics, University of Warsaw

PUBLISHED BY CAMBRIDGE UNIVERSITY PRESS (VIRTUAL PUBLISHING) FOR AND ON BEHALF OF THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGEThe Pitt Building, Trumpington Street, Cambridge CB2 IRP 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia

http://www.cambridge.org

© Cambridge University Press 1987, 2000 This edition © Cambridge University Press (Virtual Publishing) 2003

First published in printed format 1987 Second edition 2000

A catalogue record for the original printed book is available from the British Library and from the Library of Congress Original ISBN 0 521 47245 8 hardback Original ISBN 0 521 47816 2 paperback (Second edition)

ISBN 0 511 01746 4 virtual (netLibrary Edition)

In memory of Osterns – my mother’s family

Contents

Preface to the First Edition page xviiPreface to the Second Edition xviii

0 Introduction 10.1 Gauge invariance 10.2 Reasons for gauge theories of strong and electroweak interactions 3

QCD 3Electroweak theory 5

1 Classical fields, symmetries and their breaking 111.1 The action, equations of motion, symmetries and conservation laws 12

Equations of motion 12Global symmetries 13Space-time transformations 16Examples 18

1.2 Classical field equations 20Scalar field theory and spontaneous breaking of global symmetries 20Spinor fields 22

1.3 Gauge field theories 29U (1) gauge symmetry 29Non-abelian gauge symmetry 31

1.4 From classical to quantum fields (canonical quantization) 35Scalar fields 36The Feynman propagator 39Spinor fields 40Symmetry transformations for quantum fields 45

1.5 Discrete symmetries 48Space reflection 48Time reversal 53Charge conjugation 56Summary and the CPT transformation 62CP violation in the neutral K0–K0-system 64Problems 68

ix

x Contents

2 Path integral formulation of quantum field theory 712.1 Path integrals in quantum mechanics 71

Transition matrix elements as path integrals 71Matrix elements of position operators 75

2.2 Vacuum-to-vacuum transitions and the imaginary time formalism 76General discussion 76Harmonic oscillator 78Euclidean Green’s functions 82

2.3 Path integral formulation of quantum field theory 83Green’s functions as path integrals 83Action quadratic in fields 87Gaussian integration 88

2.4 Introduction to perturbation theory 90Perturbation theory and the generating functional 90Wick’s theorem 92An example: four-point Green’s function in λ�4 93Momentum space 97

2.5 Path integrals for fermions; Grassmann algebra 100Anticommuting c-numbers 100Dirac propagator 102

2.6 Generating functionals for Green’s functions and proper vertices; effectivepotential 105Classification of Green’s functions and generating functionals 105Effective action 107Spontaneous symmetry breaking and effective action 109Effective potential 111

2.7 Green’s functions and the scattering operator 113Problems 120

3 Feynman rules for Yang–Mills theories 1243.1 The Faddeev–Popov determinant 124

Gauge invariance and the path integral 124Faddeev–Popov determinant 126Examples 129Non-covariant gauges 132

3.2 Feynman rules for QCD 133Calculation of the Faddeev–Popov determinant 133Feynman rules 135

3.3 Unitarity, ghosts, Becchi–Rouet–Stora transformation 140Unitarity and ghosts 140BRS and anti-BRS symmetry 143Problems 147

4 Introduction to the theory of renormalization 1484.1 Physical sense of renormalization and its arbitrariness 148

Bare and ‘physical’ quantities 148Counterterms and the renormalization conditions 152

Contents xi

Arbitrariness of renormalization 153Final remarks 156

4.2 Classification of the divergent diagrams 157Structure of the UV divergences by momentum power counting 157Classification of divergent diagrams 159Necessary counterterms 161

4.3 λ�4: low order renormalization 164Feynman rules including counterterms 164Calculation of Fig. 4.8(b) 166Comments on analytic continuation to n �= 4 dimensions 168Lowest order renormalization 170

4.4 Effective field theories 173Problems 175

5 Quantum electrodynamics 1775.1 Ward–Takahashi identities 179

General derivation by the functional technique 179Examples 181

5.2 Lowest order QED radiative corrections by the dimensional regularizationtechnique 184General introduction 184Vacuum polarization 185Electron self-energy correction 187Electron self-energy: IR singularities regularized by photon mass 190On-shell vertex correction 191

5.3 Massless QED 1945.4 Dispersion calculation of O(α) virtual corrections in massless QED, in

(4 ∓ ε) dimensions 196Self-energy calculation 197Vertex calculation 198

5.5 Coulomb scattering and the IR problem 200Corrections of order α 200IR problem to all orders in α 205Problems 208

6 Renormalization group 2096.1 Renormalization group equation (RGE) 209

Derivation of the RGE 209Solving the RGE 212Green’s functions for rescaled momenta 214RGE in QED 215

6.2 Calculation of the renormalization group functions β, γ, γm 2166.3 Fixed points; effective coupling constant 219

Fixed points 219Effective coupling constant 222

6.4 Renormalization scheme and gauge dependence of the RGEparameters 224

xii Contents

Renormalization scheme dependence 224Effective α in QED 226Gauge dependence of the β-function 227Problems 228

7 Scale invariance and operator product expansion 2307.1 Scale invariance 230

Scale transformations 230Dilatation current 233Conformal transformations 235

7.2 Broken scale invariance 237General discussion 237Anomalous breaking of scale invariance 238

7.3 Dimensional transmutation 2427.4 Operator product expansion (OPE) 243

Short distance expansion 243Light-cone expansion 247

7.5 The relevance of the light-cone 249Electron–positron annihilation 249Deep inelastic hadron leptoproduction 250Wilson coefficients and moments of the structure function 254

7.6 Renormalization group and OPE 256Renormalization of local composite operators 256RGE for Wilson coefficients 259OPE beyond perturbation theory 261

7.7 OPE and effective field theories 262Problems 269

8 Quantum chromodynamics 2728.1 General introduction 272

Renormalization and BRS invariance; counterterms 272Asymptotic freedom of QCD 274The Slavnov–Taylor identities 277

8.2 The background field method 2798.3 The structure of the vacuum in non-abelian gauge theories 282

Homotopy classes and topological vacua 282Physical vacuum 284�-vacuum and the functional integral formalism 287

8.4 Perturbative QCD and hard collisions 290Parton picture 290Factorization theorem 291

8.5 Deep inelastic electron–nucleon scattering in first order QCD (Feynmangauge) 293Structure functions and Born approximation 293Deep inelastic quark structure functions in the first order in the strongcoupling constant 298Final result for the quark structure functions 302

Contents xiii

Hadron structure functions; probabilistic interpretation 3048.6 Light-cone variables, light-like gauge 3068.7 Beyond the one-loop approximation 312

Comments on the IR problem in QCD 314Problems 315

9 Chiral symmetry; spontaneous symmetry breaking 3179.1 Chiral symmetry of the QCD lagrangian 3179.2 Hypothesis of spontaneous chiral symmetry breaking in strong interactions 3209.3 Phenomenological chirally symmetric model of the strong interactions

(σ -model) 3249.4 Goldstone bosons as eigenvectors of the mass matrix and poles of Green’s

functions in theories with elementary scalars 327Goldstone bosons as eigenvectors of the mass matrix 327General proof of Goldstone’s theorem 330

9.5 Patterns of spontaneous symmetry breaking 3339.6 Goldstone bosons in QCD 337

10 Spontaneous and explicit global symmetry breaking 34210.1 Internal symmetries and Ward identities 342

Preliminaries 342Ward identities from the path integral 344Comparison with the operator language 347Ward identities and short-distance singularities of the operatorproducts 348Renormalization of currents 351

10.2 Quark masses and chiral perturbation theory 353Simple approach 353Approach based on use of the Ward identity 354

10.3 Dashen’s theorems 356Formulation of Dashen’s theorems 356Dashen’s conditions and global symmetry broken by weak gauge interactions 358

10.4 Electromagnetic π+–π0 mass difference and spectral function sumrules 362Electromagnetic π+–π0 mass difference from Dashen’s formula 362Spectral function sum rules 363Results 366

11 Higgs mechanism in gauge theories 36911.1 Higgs mechanism 36911.2 Spontaneous gauge symmetry breaking by radiative corrections 37311.3 Dynamical breaking of gauge symmetries and vacuum alignment 379

Dynamical breaking of gauge symmetry 379Examples 382Problems 388

12 Standard electroweak theory 38912.1 The lagrangian 391

xiv Contents

12.2 Electroweak currents and physical gauge boson fields 39412.3 Fermion masses and mixing 39812.4 Phenomenology of the tree level lagrangian 402

Effective four-fermion interactions 403Z0 couplings 406

12.5 Beyond tree level 407Renormalization and counterterms 407Corrections to gauge boson propagators 411Fermion self-energies 418Running α(µ) in the electroweak theory 419Muon decay in the one-loop approximation 422Corrections to the Z0 partial decay widths 430

12.6 Effective low energy theory for electroweak processes 435QED as the effective low energy theory 438

12.7 Flavour changing neutral-current processes 441QCD corrections to CP violation in the neutral kaon system 445Problems 456

13 Chiral anomalies 45713.1 Triangle diagram and different renormalization conditions 457

Introduction 457Calculation of the triangle amplitude 459Different renormalization constraints for the triangle amplitude 464Important comments 465

13.2 Some physical consequences of the chiral anomalies 469Chiral invariance in spinor electrodynamics 469π0 → 2γ 471Chiral anomaly for the axial U (1) current in QCD; UA(1) problem 473Anomaly cancellation in the SU (2)×U (1) electroweak theory 475Anomaly-free models 478

13.3 Anomalies and the path integral 478Introduction 478Abelian anomaly 480Non-abelian anomaly and gauge invariance 481Consistent and covariant anomaly 484

13.4 Anomalies from the path integral in Euclidean space 486Introduction 486Abelian anomaly with Dirac fermions 488Non-abelian anomaly and chiral fermions 491Problems 492

14 Effective lagrangians 49514.1 Non-linear realization of the symmetry group 495

Non-linear σ -model 495Effective lagrangian in the ξa(x) basis 500Matrix representation for Goldstone boson fields 502

14.2 Effective lagrangians and anomalies 504

Contents xv

Abelian anomaly 505The Wess–Zumino term 506Problems 508

15 Introduction to supersymmetry 50915.1 Introduction 50915.2 The supersymmetry algebra 51115.3 Simple consequences of the supersymmetry algebra 51315.4 Superspace and superfields for N = 1 supersymmetry 515

Superspace 515Superfields 519

15.5 Supersymmetric lagrangian; Wess–Zumino model 52115.6 Supersymmetry breaking 52415.7 Supergraphs and the non-renormalization theorem 531

Appendix A: Spinors and their properties 539Lorentz transformations and two-dimensional representations of thegroup SL(2,C) 539Solutions of the free Weyl and Dirac equations and their properties 546Parity 550Time reversal 551Charge conjugation 552

Appendix B: Feynman rules for QED and QCD and Feynman integrals 555Feynman rules for the λ�4 theory 555Feynman rules for QED 556Feynman rules for QCD 557Dirac algebra in n dimensions 558Feynman parameters 559Feynman integrals in n dimensions 559Gaussian integrals 560λ-parameter integrals 560Feynman integrals in light-like gauge n · A = 0, n2 = 0 561Convention for the logarithm 561Spence functions 562

Appendix C: Feynman rules for the Standard Model 563Propagators of fermions 563Propagators of the gauge bosons 564Propagators of the Higgs and Goldstone bosons 565Propagators of the ghost fields 566Mixed propagators (only counterterms exist) 567Gauge interactions of fermions 567Yukawa interactions of fermions 570Gauge interactions of the gauge bosons 571Self-interactions of the Higgs and Goldstone bosons 573Gauge interactions of the Higgs and Goldstone bosons 574Gauge interactions of the ghost fields 578Interactions of ghosts with Higgs and Goldstone bosons 579

xvi Contents

Appendix D: One-loop Feynman integrals 583Two-point functions 583Three- and four-point functions 585General expressions for the one-loop vector boson self-energies 586

Appendix E: Elements of group theory 591Definitions 591Transformation of operators 593Complex and real representations 593Traces 594σ -model 596

References 599Index 605

Preface to the First Edition

This book has its origin in a long series of lectures given at the Institute forTheoretical Physics, Warsaw University. It is addressed to graduate students andto young research workers in theoretical physics who have some knowledge ofquantum field theory in its canonical formulation, for instance at the level of twovolumes by Bjorken & Drell (1964, 1965). The book is intended to be a relativelyconcise reference to some of the field theoretical tools used in contemporaryresearch in the theory of fundamental interactions. It is a technical book andnot easy reading. Physical problems are discussed only as illustrations of certaintheoretical ideas and of computational methods. No attempt has been made toreview systematically the present status of the theory of fundamental interactions.

I am grateful to Wojciech Krolikowski, Maurice Jacob and Peter Landshofffor their interest in this work and strong encouragement. My warm thanksgo to Antonio Bassetto, Wilfried Buchmuller, Wojciech Krolikowski, HeinrichLeutwyler, Peter Minkowski, Olivier Piguet, Jacek Prentki, Marco Roncadelli,Henri Ruegg and Wojtek Zakrzewski for reading various chapters of this book andfor many useful comments, and especially to Peter Landshoff for reading most ofthe preliminary manuscript.

I am also grateful to several of my younger colleagues at the Institute forTheoretical Physics in Warsaw for their stimulating interactions. My thanks goto Andrzej Czechowski for his collaboration at the early stage of this projectand for numerous useful discussions. I am grateful to Wojciech Debski, MarekOlechowski, Jacek Pawełczyk, Andrzej Turski, Robert Budzynski, KrzysztofMeissner and Michał Spalinski, and particularly to Paweł Krawczyk for checkinga large part of the calculations contained in this book.

Finally my thanks go to Zofia Ziołkowska for her contribution to the preparationof the manuscript.

Stefan PokorskiWarsaw, 1985

xvii

Preface to the Second Edition

This new edition offers a substantial extension of topics covered by the book.The main additions are Chapter 1 and Chapter 12 and extended Appendices.Chapter 1 makes the book more self-contained. It gives a systematic introductionto classical field theories and a brief discussion of their canonical quantization, assome intuition based on canonical quantization proves to be very useful even if themain emphasis is on the path integral approach. Also in Chapter 1 the reader canfind a thorough discussion of discrete symmetries C , P and T .

Chapter 12 gives a concise but systematic and self-contained introduction to theelectroweak theory. This is an important completion for a modern book on quantumfield theory and fundamental interactions, which was missing in the first edition.Appendices A, C and D are new. In particular, Appendix C contains the completeset of Feynman rules for the Standard Model, including counterterms, which is noteasily available in the literature. The new Appendix A is a substantial extension ofthe previous Appendix C. Several smaller changes and corrections have been madein a number of places in the text. An important addition is Section 7.7, in whichthe modern approach to effective field theories is presented.

I hope that the additions leave intact the main feature of the book: it is still noteasy reading!

I am grateful to many people for their help in the completion of this secondedition. Very special thanks go to Piotr Chankowski. His help and collaboration inwriting Chapters 1 and 12 and the Appendices was absolutely invaluable. Thanks,Piotr. I also thank Mikolaj Misiak and Janusz Rosiek for their collaboration onSections 7.7 and 15.6, respectively.

I am deeply indebted to Howie Haber for his careful reading of a large part of thenew material. I am grateful to Zygmunt Ajduk, Ratindranath Akhoury, RiccardoGuida, Krzysztof Meissner, Marek Olechowski, Jacek Pawełczyk, Carlos Savoyand Kay Wiese for their numerous comments and corrections. I would also like to

xviii

Preface to the Second Edition xix

thank Peter Landshoff for his constant encouragement and patience during quite afew years before this edition finally materialized.

Comments and corrections are welcome by e-mail at [email protected] will be made accessible at the www page: http://www.fuw.edu.pl/∼pokorski/

Stefan PokorskiWarsaw, 1999

0

Introduction

0.1 Gauge invariance

It seems appropriate to begin this book by quoting the following experimentalinformation (Review of Particle Properties 1996):

electron life-time > 4.3 × 1023 years (68% C.L.)neutron life-time for the electric charge

non-conserving decays (n → p + neutrals) � 1019 yearsproton life-time > 1032 yearsphoton mass < 6 × 10−22 MeVneutrino (νe) mass < 15 × 10−6 MeV

From the first three lines of this table we see that the conservation of the electriccharge is proved experimentally much more poorly than the conservation of thebaryonic charge. Nevertheless, nobody is seriously contesting electric chargeconservation whereas experiments searching for proton decay belong to the presentfrontiers of physics. The reason for this lies in the general conviction that thetheory of electromagnetism has gauge symmetry† whereas no gauge invarianceprinciple can be invoked to protect baryonic charge conservation. Exact gaugeinvariance protects the conservation of the electric charge. It also implies themasslessness of the photon and as seen from the table the present experimentallimit on the photon mass is indeed many orders of magnitude better than for theother ‘massless’ particle: the electron neutrino.

It should be stressed at this point that U (1) gauge invariance implies globalU (1) invariance but, of course, the opposite is not true. A tiny mass for the photonwould destroy the gauge invariance of electrodynamics but leave unaffected all itsEarth-bound effects, including the quantum ones, as long as the electric chargewas conserved. In particular, such a theory is also renormalizable (Matthews 1949,

† The reader who is not familiar with notions of global symmetry and gauge symmetry is advised to readChapter 1 first.

1

2 0 Introduction

Boulware 1970, Salam & Strathdee 1970) since longitudinally polarized photonsdecouple from the conserved current.

Thus one may ask what are the virtues of gauge invariance? We trust electriccharge conservation, because we expect that gauge invariance is behind it, and wedoubt baryon charge conservation, though this has been much better proved exper-imentally than electric charge conservation. We do not trust global symmetries ascandidates for underlying first principles! A global symmetry gives us a freedomof convention: choice of a reference frame (the phase of the electron wave-functionfor the U (1) symmetry of electrodynamics). It can be redefined freely, providedthat all observers in the universe redefine it in exactly the same way. This soundsunphysical and we are led to propose that this freedom of convention is presentindependently at every space-time point or is not present at all as an exact law ofnature. (Approximate global symmetries may, nevertheless, be and are very usefulin describing the fundamental interactions.) This aesthetical argument may notconvince everybody. Those who remain sceptical should then remember that gaugetheories give an economical description of the laws of nature based on well-definedunderlying principles which has been phenomenologically successful. As we knowat present, this statement accounts not only for electrodynamics with its U (1)abelian gauge symmetry, but also for weak and strong interactions successfullydescribed by gauge field theories with non-abelian symmetry groups: SU (2) ×U (1) and SU (3), respectively. And non-abelian gauge symmetries are morerestrictive and more profound than the U (1) symmetry. In particular, non-abeliangauge bosons carry the group charges and their mass terms in the lagrangianwould in general, unless introduced by spontaneous symmetry breaking, destroynot only the gauge symmetry but also the current conservation and therefore therenormalizability of the theory. The standard experimental evidence for gaugetheories of weak and strong interactions is briefly summarized in the next section.

We end this section with a short historical ‘footnote’ (Pauli 1933). Theterminology ‘gauge invariance’ can be traced back to Weyl’s studies (Weyl 1919)of invariance under space-time-dependent changes of gauge (scale) in an attempt tounify gravity and electromagnetism. This attempt proved, however, unsuccessful.In 1926, Fock observed (Fock 1926) that one could base quantum electrodynamics(QED) of scalar particles on the operator

−ih∂

∂xµ− e

cAµ

where Aµ is the electromagnetic four-potential and that the equations were invari-ant under the transformation

Aµ → Aµ + ∂ f (x)

∂xµ, � → � exp[ie f (x)/hc]

0.2 Reasons for gauge theories 3

which he called gradient transformation. London (1927) pointed out the similarityof Fock’s to Weyl’s earlier work: instead of Weyl’s scale change a local phasechange was considered by Fock. In 1929, Weyl studied invariance under this phasechange but he kept unchanged his earlier terminology ‘gauge invariance’ (Weyl1929). The concept of gauge transformations was generalized to non-abelian gaugegroups by Yang & Mills (1954). Similar ideas were also proposed much earlier byKlein (1939) and by Shaw (1955).

0.2 Reasons for gauge theories of strong and electroweak interactions

We summarize very briefly the standard arguments in favour of gauge theoriesin elementary particle physics. Both quantum chromodynamics (QCD) and theGlashow–Salam–Weinberg theory are syntheses of our understanding of funda-mental interactions progressing over many past years.

QCD

QCD emerged as a development of the Gell-Mann–Zweig quark model for hadrons(Gell-Mann 1964, Zweig 1964). The latter was postulated as a rationale for thesuccessful SU (3) classification of hadrons (today one should say flavour SU (3)).Assigning quarks q to the fundamental representation of SU (3), not realized by anyknown hadrons, and giving them spin one-half one obtains the phenomenologicallysuccessful SU (3) and SU (6) schemes. SU (6) is obtained by adjoining the groupSU (2) of spin rotation to the internal symmetry group SU (3) for baryons (qqq)and mesons (qq). In particular, the known hadrons indeed realize only thoserepresentations of SU (3) which are given by the composite model. The quarkmodel for hadrons, successful as it was, appeared, however, to have difficultiesin reconciling the Fermi statistics for quarks with the most natural assumptionthat in the lowest-lying hadronic states all the relative angular momenta amongconstituent quarks vanish (s-wave states). Thus, baryon wave-functions should beantisymmetric in spin and flavour degrees of freedom. This is not the case in theoriginal quark model as can be immediately seen from inspection of the ++( 3

2+)

wave-function which must be u↑u↑u↑; u denotes the quark with electric chargeQ = 3

2 , the arrow denotes spin Sz = 12 for each quark.

The difficulty can be resolved by postulating a new internal quantum numberfor quarks which has been called colour (Greenberg 1964, Han & Nambu 1965,Nambu 1966, Bardeen, Fritzsch & Gell-Mann 1973). If a quark of each flavourhas three, otherwise indistinguishable, colour states, Fermi statistics is saved byusing a totally antisymmetric colour wave-function εabcua↑ub↑uc↑. Assumingfurthermore that (i) strong interactions are invariant under global SU (3)colour

4 0 Introduction

Fig. 0.1. The process e+e− → γ → hadrons in the parton model. The sum is taken overall hadronic states in the reaction e+e− → γ → qq → hadrons.

transformations (the states may then be classified by their SU (3)colour representa-tion) and (ii) physical hadrons are colourless, i.e. they are singlets under SU (3)colour

(quark confinement), we can understand why only qqq and qq states, and not qq orqqqq etc., exist in nature: the singlet representation appears only in the 3 × 3 × 3and 3 × 3 products.

The concept of colour is also supported by at least two other, strong arguments.One is based on the parton model (Feynman 1972) approach to the reactione+e− → hadrons. The total cross section for this process is then given by thediagram in Fig. 0.1 and the ratio

R = σ(e+e− → hadrons)

σ (e+e− → µ+µ−)(0.1)

is predicted to be

R = e2 ∑q Q2

q

e2=∑

q

Q2q = 3 × (

49 + 1

9 + 49 + 1

9 + 19 + · · ·)

= 113 (including quarks up to b) (0.2)

The experimental value of R is in good agreement with this prediction and in pooragreement with the colourless prediction 11

9 .Yet another reason for colour is provided by the decay π0 → 2γ . Here again the

number of quark states matters in explaining the width of π0 → 2γ . This problemwill be discussed in more detail in Chapter 13.

The concept of colour certainly underlies what we believe to be the true theoryof strong interactions, namely QCD. However, the theory also has several otherbasic features which are partly suggested by experimental observations and partlyrequired by theoretical consistency. Firstly, it is assumed that strong interactionsact on the colour quantum numbers and only on them. Experimentally there is noevidence for any flavour dependence of strong forces; all flavour-dependent effectscan be explained by quark mass differences and the origin of the quark masses,though not satisfactorily understood yet, is expected to be outside of QCD. Inaddition, only colour symmetry can be assumed to be an exact symmetry (flavour


symmetry is evidently broken) and this, combined with the assumption that itis a gauge symmetry (Han & Nambu 1965, Fritzsch, Gell-Mann & Leutwyler1973), has profound implications: asymptotic freedom (Gross & Wilczek 1973,Politzer 1973) and presumably, though not proven, confinement of quarks. Both arewelcome features. Asymptotic freedom means that the forces become negligibleat short distances and consequently the interaction between quarks by exchangeof non-abelian gauge fields (gluons) is consistent with the successful, as the firstapproximation, description of the deep inelastic scattering in the framework ofthe parton model. It has been shown that only non-abelian gauge theories areasymptotically free (Coleman & Gross 1973). Confinement of the colour quantumnumbers, i.e. of quarks and gluons, has not yet been proved to follow from QCD butit is likely to be true, reflecting strongly singular structure of the non-abelian gaugetheory in the IR region. Once we assume colourful quarks as elementary objectsin hadrons, confinement of colour is desirable in view of the so far unsuccessfulexperimental search for free quarks and to avoid a proliferation of unwanted states.

An important line of argument in favour of gluons as vector bosons beginswith approximate chiral symmetry of strong interactions (Chapter 9). Coupling offermions with vector and axial-vector fields, but not with scalars or pseudoscalars,is chirally invariant. A theory with axial-vector gluons based on the SU (3) groupcannot be consistently renormalized because of anomalies (Chapter 13). Thus wearrive at vector gluon interaction.

In recent years there has been a lot of research in QCD perturbation theoryneglecting the unsolved confinement problem. Of course, this cannot be fullysatisfactory since experimenters collide hadrons and not quarks and gluons. Onecan nevertheless argue that it is a justifiable approximation at short distances.Thus, at its present stage the theory provides us with calculable corrections to thefree-field behaviour of quarks and gluons in the parton model and can be testedin the deep inelastic region. Given the accuracy of the calculations and of theexperimental data one cannot claim yet to have strongly positive experimentalverification of perturbative QCD predictions. However, experimental results arecertainly consistent with QCD (in particular jet physics already provides us withgood evidence for the vector nature of gluons) and in view of its elegance andself-consistency there are few sceptical of its chance of being the theory of stronginteractions.

Electroweak theory

There is, at present, impressive experimental evidence for the electroweak gaugetheory with the gauge symmetry spontaneously broken. To introduce the Glashow–Salam–Weinberg theory we recall first that the effective Fermi lagrangian for the

6 0 Introduction

charged-current weak interactions, valid at low energies, has conventionally beentaken to be (for a systematic account of the weak interactions phenomenology see,for example, Gasiorowicz (1966) and more recently Abers & Lee (1973) and Taylor(1976))

Leff(x) = −2√

2 Gµ J †µ(x)Jµ(x) (0.3)

where the Fermi β-decay constant Gµ = 1.165 × 10−5 GeV−2 (h = c = 1) andthe charged current Jµ(x) is composed of several pieces, each with V–A structure.In terms of the lepton and quark fields it can be written as follows

Jµ =∑

f

�f

L γ µT−� fL (0.4)

with

T± = 12(σ

1 ± σ 2) = T 1 ± iT 2

where σ a are Pauli matrices and for weak interactions, as known in the 1960s,

�f

L = 12(1 − γ5)

{(νe

e−

),

(νµ

µ−

),

(ud ′

)}(0.5)

The subscript L stands for the left-handed fermions. The prime superscriptindicates the existence of mixing of the quark fields observed in strong interactions(mass eigenstates):

d ′ = d cos�C + s sin�C

where the angle �C is known as the Cabibbo angle and has been measured in weakdecays of strange particles.

We now extend the Fermi–Cabibbo theory by several additional assumptions.Firstly, we postulate the existence of a symmetry group, global for the time being,for the weak interactions. Having the current (0.4) it is natural to postulate SU (2)symmetry and consequently the existence of the neutral current corresponding tothe third generator of the SU (2)

[T+, T−] = 2T 3 (0.6)

which would induce transitions, such as, for instance, those in Fig. 0.2 occurringwith a similar strength to the charged-current reactions. Neutral-current weaktransitions with the expected strength have been discovered at CERN (Hasertet al. 1973). However, they (i) are not of purely V–A character as expected fromthe SU (2) model, and (ii) always conserve strangeness to a very good accuracy.According to the existing experimental limits the strangeness non-conservingneutral-current transitions (like K0

L → µ+µ− or K0 ↔ K0) are suppressed by manyorders of magnitude as compared to the standard weak processes. Both factors


Fig. 0.2. Some neutral-current weak transitions.

call for further invention in searching for a realistic theory of weak interactions.Glashow, Iliopoulos & Maiani (1970) have discovered that the problem of thestrangeness non-conserving neutral current is solved if the set of fermionic doublets�

fL is completed with a fourth one(

cs ′

), s ′ = −d sin�C + s cos�C

One can immediately check that with the s ′ orthogonal to d ′ the neutral current isdiagonal in flavour. Thus, they have predicted the existence of the charm quarkdiscovered later at SLAC (Aubert et al. 1974, Augustin et al. 1974). Also thedoublet classification of the left-handed fermions with an equal number of leptonand quark doublets, now further confirmed by the experimental discovery of the(

ντ

τ

)and

(tb

)doublets,† has emerged as an important property of weak interactions. This hasprofound implications for a successful extension of the effective model into a non-abelian local gauge field theory: it ensures the cancellation of the chiral anomalyand consequently the renormalizability of the theory. A highly consistent schemebegins to be expected.

The next step towards the final form of the Glashow–Salam–Weinberg theory is a‘unification’ of weak and electromagnetic interactions (Schwinger 1957, Glashow1961, Salam & Ward 1964). Thus, we want the electric charge Q also to bethe generator of the symmetry group of our theory. To achieve this in the mosteconomical way we notice that for the left-handed doublets of fermions we can

† The t quark was discovered by the CDF and D0 groups in Fermilab in 1994.

8 0 Introduction

define a new quantum number Y (weak hypercharge)

Y = Q − T 3 (0.7)

such that each doublet of the left-handed fermions is an eigenvector of the operatorY (e.g. YνL = YeL = − 1

2). Therefore Y commutes with the generators of SU (2) andincluding the right-handed fermions by the prescription Y = Q (they are singletswith respect to SU (2)) we arrive at the SU (2) × U (1) symmetry group for theelectroweak interactions. The U (1) current reads

Jµ

Y =∑

f

�f

L γ µY�f

L +∑

f

�f

Rγ µY�f

R (0.8)

where �f

L(R) are left- (right-)handed chiral lepton and quark fields. According to(0.7), the electromagnetic current can be written as follows

Jµem = Jµ

Y +∑

f

�f

L γ µT 3�f

L

=∑

f

� f Qγ µ� f (0.9)

The SU (2) × U (1) group is the minimal one which contains the electromagneticand weak currents. With electromagnetism being described by a gauge field Aµ

our minimal model ‘unifying’ electromagnetic and weak interactions requires theYang–Mills gauge fields W α

µ and Bµ to couple to the SU (2)×U (1) currents givingthe interaction

g J αµ W αµ + g′ Jµ

Y Bµ (0.10)

Since we are dealing with a direct group product the couplings g and g′ areindependent. Therefore the significance of the electroweak unification comes fromthe choice of a minimal group structure which takes account of both currents.Even with two independent coupling constants, this ‘unification’ has a predictivepower: it determines the structure of the parity violation in the neutral weak current(see (0.15) below) and gives the W 3–γ mixing in terms of g′ and g, which as wewill see, reduces the number of free parameters of the theory.

The charged gauge bosons

W±µ = 1√

2(W 1

µ ± iW 2µ) (0.11)

couple directly to the experimentally observed weak currents. As follows from theformula

Q = 1

g(gT 3)+ 1

g′(g′Y ) (0.12)


the photon field Aµ (determined by the fact that it couples to the electromagneticcurrent) must be a combination of the W 3 and B bosons

Aµ =(

1

g2+ 1

g′2

)−1/2(1

gW 3

µ +1

g′Bµ

)(0.13)

The orthogonal combination

Zµ =(

1

g2+ 1

g′2

)−1/2( 1

g′W 3

µ −1

gBµ

)(0.14)

couples to the neutral weak current J zµ

J zµ =

g

cos�W(J 3

µ − sin2 �W J emµ ) (0.15)

where Jµem is given by (0.9) and we have introduced the standard notation in terms

of the so-called Weinberg angle (Glashow 1961, Weinberg 1967b)

sin2 �W = g′2

g2 + g′2or tan�W = g′

gor e = g sin�W = g′ cos�W

(0.16)(the last relation follows from the fact that the electromagnetic coupling constantis electric charge e, e > 0).

If initially all vector fields are massless, we must now introduce masses forintermediate vector boson fields W± and Z to account for the weakness ofthe weak interactions as compared to electromagnetism. One knows that themassive Yang–Mills theory, although less divergent than a theory with arbitrarycouplings, is not renormalizable unless the vector boson masses are introducedby means of the Higgs mechanism which breaks the non-abelian gauge invariancespontaneously (’t Hooft 1971a, b). In the minimal model this is achieved (Weinberg1967b, Salam 1968) with one complex SU (2) doublet of scalar fields whose neutralcomponent has a non-vanishing vacuum expectation value v. Then the bosonmasses are (at the tree level)

MW = 12vg, MZ = 1

2v(g′2 + g2)1/2

so that

MW/MZ cos�W = 1 (0.17)

Thus, apart from the fermion mass matrix and the scalar self-coupling we have athree-parameter (g, g′, v) electroweak theory. For instance, knowing the electro-magnetic coupling α = e2/4π , the Fermi constant Gµ and the Weinberg angle(parity non-conservation in all neutral-current transitions is consistent with the

10 0 Introduction

form (0.15) and with sin2 �W ≈ 0.23 one predicts, at the tree level,

MW = MZ cos�W =(

πα√2Gµ

)1/2 1

sin�W≈ 80 GeV (0.18)

Vector bosons with the expected masses were discovered at CERN in 1982!This triumph of the spontaneously broken Yang–Mills theory has since beenconfirmed by comparing higher accuracy data with higher order calculations.However, the discovery of the Higgs particle still remains (in 1999) a challengefor future experiments. We would like to stress the role of gauge symmetryand the unification idea, apparently coupled to each other, in this very successfuldescription of the electroweak interactions.

Non-abelian gauge theories are considered at present to be the most promisingtheoretical framework for fundamental interactions. They are also extensivelystudied in the hope of providing the unified description of all interactions (grandunification, supergravity) going beyond the so-called standard SU (3) × SU (2) ×U (1) model summarized in this section.

1

Classical fields, symmetries and their breaking

Classically, we distinguish particles and forces which are responsible for interac-tion between particles. The forces are described by classical fields. The motionof particles in force fields is subject to the laws of classical mechanics. Quanti-zation converts classical mechanics into quantum mechanics which describes thebehaviour of particles at the quantum level. A state of a particle is describedby a vector |�(t)〉 in the Hilbert space or by its concrete representation, e.g.the wave-function �(x) = 〈x|�(t)〉 (x = (t, x)) whose modulus squared isinterpreted as the density of probability of finding the particle in point x at thetime t . As a next step, the wave-function is interpreted as a physical field andquantized. We then arrive at quantum field theory as the universal physicalscheme for fundamental interactions. It is also reached by quantization of fieldsdescribing classical forces, such as electromagnetic forces. The basic physicalconcept which underlies quantum field theory is the equivalence of particles andforces. This logical structure of theories for fundamental interactions is illustratedby the diagram shown below:

PARTICLES FORCESClassical mechanics Classical field theory

(quantization) (quantization)

Quantum mechanics(wave-function �(x) is interpretedas a physical field and quantized)

QUANTUM FIELD THEORY

This chapter is devoted to a brief summary of classical field theory. The readershould not be surprised by such formulations as, for example, the ‘classical’ Diracfield which describes the spin 1

2 particle. It is interpreted as a wave function for theDirac particle. Our ultimate goal is quantum field theory and our classical fields in

11

12 1 Classical fields, symmetries and their breaking

this chapter are not necessarily only the fields which describe the classical forcesobserved in Nature.

1.1 The action, equations of motion, symmetries and conservation laws

Equations of motion

All fundamental laws of physics can be understood in terms of a mathematicalconstruct: the action. An ansatz for the action S = ∫

dt L = ∫d4x L can be

regarded as a formulation of a theory. In classical field theory the lagrangiandensity L is a function of fields � and their derivatives. In general, the fields �

are multiplets under Lorentz transformations and in a space of internal degreesof freedom. It is our experience so far that in physically relevant theoriesthe action satisfies several general principles such as: (i) Poincare invariance(or general covariance for theories which take gravity into consideration), (ii)locality of L and its, at most, bilinearity in the derivatives ∂ν�(x) (to get atmost second order differential equations of motion),† (iii) invariance under allsymmetry transformations which characterize the considered physical system, (iv)S has to be real to account for the absence of absorption in classical physics andfor conservation of probability in quantum physics. The action is then the mostgeneral functional which satisfies the above constraints.

From the action we get:

equations of motion (invoking Hamilton’s principle);conservations laws (from Noether’s theorem);transition from classical to quantum physics (by using path integrals or canonicalquantization).

In this section we briefly recall the first two results. Path integrals are discussed indetail in Chapter 2.

The equations of motion for a system described by the action

S =∫ t1

t0

dt∫

Vd3x L

(�(x), ∂µ�(x)

)(1.1)

(the volume V may be finite or infinite; in the latter case we assume the systemto be localized in space‡) are obtained from Hamilton’s principle. This ansatz(whose physical sense becomes clearer when classical theory is considered as alimit of quantum theory formulated in terms of path integrals) says: the dynamicsof the system evolving from the initial state �(t0, x) to the final state �(t1, x) is

† Later we shall encounter effective lagrangians that may contain terms with more derivatives. Such terms canusually be interpreted as a perturbation.

‡ Although we shall often use plane wave solutions to the equations of motion, we always assume implicitlythat real physical systems are described by wave packets localized in space.

1.1 Action, equations of motion, symmetries and conservation 13

such that the action (taken as a functional of the fields and their first derivatives)remains stationary during the evolution, i.e.

δS = 0 (1.2)

for arbitrary variations

�(x) → �(x)+ δ�(x) (1.3)

which vanish on the boundary of � ≡ ( t, V ).Explicit calculation of the variation δS gives:

δS =∫�

d4x

[∂L∂�

δ�+ ∂L∂(∂ν�)

δ(∂ν�)

](1.4)

(summation over all fields � and for each field � over its Lorentz and ‘internal’indices is always understood). Since in the variation the coordinates x do notchange we have:

δ(∂ν�) = ∂νδ� (1.5)

and (1.4) can be rewritten as follows:

δS =∫�

d4x

{[∂L∂�

− ∂ν∂L

∂(∂ν�)

]δ�+ ∂ν

[∂L

∂(∂ν�)δ�

]}(1.6)

The second term in (1.6) is a surface term which vanishes and, therefore, thecondition δS = 0 gives us the Euler–Lagrange equations of motion for the classicalfields:

∂L∂�

µ

i

− ∂ν∂L

∂(∂ν�µ

i )= 0

ν, µ = 0, 1, 2, 3

i = 1, . . . , n(1.7)

(here we keep the indices explicitly, with � taken as a Lorentz vector; is areinternal quantum number indices). It is important to notice that lagrangian densitieswhich differ from each other by a total derivative of an arbitrary function of fields

L′ = L+ ∂µ�µ(�) (1.8)

give the same classical equations of motion (due to the vanishing of variations offields on the boundary of �).

Global symmetries

Eq. (1.6) can also be used to derive conservation laws which follow from a certainclass of symmetries of the physical system. Let us consider a Lie group ofcontinuous global (x-independent) infinitesimal transformations (here we restrictourselves to unitary transformations; conformal transformations are discussed in


Chapter 7) acting in the space of internal degrees of freedom of the fields �i .Under infinitesimal rotation of the reference frame (passive view) or of the physicalsystem (active view) in that space:

�i (x) → �′i (x) = �i (x)+ δ0�i (x) (1.9)

where

δ0�i (x) = −i�aT ai j� j (x) (1.10)

and �′i (x) is understood as the i th component of the field �(x) in the new reference

frame or of the rotated field in the original frame (active). Note that a rotation ofthe reference frame by angle � corresponds to active rotation by (−�). The T asare a set of hermitean matrices T a

i j satisfying the Lie algebra of the group G[T a, T b

] = icabcT c, Tr(T aT b) = 12δ

ab (1.11)

and the �as are x-independent.Under the change of variables � → �′(x) given by (1.9) the lagrangian density

is transformed into

L′(�′(x), ∂µ�′(x)) ≡ L

(�(x), ∂µ�(x)

)(1.12)

and, of course, the action remains numerically unchanged:

δ�S = S′ − S =∫�

d4x[L′(�′, ∂µ�′)− L(�, ∂µ�)

] = 0 (1.13)

(we use the symbol δ�S for the change of the action under transformation (1.9)on the fields to distinguish it from the variations δS′ and δS). For the variationswe have δS′ = δS. Thus, if �(x) describes a motion of a system, �′(x) is also asolution of the equations of motion for the transformed fields which, however, ingeneral are different in form from the original ones.

Transformations (1.9) are symmetry transformations for a physical system ifits equations of motion remain form-invariant in the transformed fields. In otherwords, a solution to the equations of motion after being transformed according to(1.9) remains a solution of the same equations. This is ensured if the density L isinvariant under transformations (1.9) (for scale transformations see Chapter 7):

L′(�′(x), ∂µ�′(x)) = L(�′(x), ∂µ�′(x)) (1.14)

or, equivalently,

L(�′(x), ∂µ�′(x)) = L(�(x), ∂µ�(x)).

Indeed, variation of the action generated by arbitrary variations of the fields�′(x) with boundary conditions (1.3) is again given by (1.6) (with � → �′ etc.)


and we derive the same equations for �′(x) as for �(x). Moreover, the change(1.14) of the lagrangian density under the symmetry transformations:

L′(�′, ∂µ�′)− L(�, ∂µ�) = L(�′, ∂µ�′)− L(�, ∂µ�) = 0 (1.15)

is formally given by the integrand in (1.6) with δ�s given by (1.9). It follows from(1.15) and the equations of motion that the currents

j aµ(x) = −i

∂L∂(∂µ�i )

T ai j� j (1.16)

are conserved and the charges

Qa(t) =∫

d3x ja0 (t, x) (1.17)

are constants of motion, provided the currents fall off sufficiently rapidly at thespace boundary of �.

It is very important to notice that the charges defined by (1.17) (even if theyare not conserved) are the generators of the transformation (1.10). Firstly, theysatisfy (in an obvious way) the same commutation relation as the matrices T a .Secondly, they indeed generate transformations (1.10) through Poisson brackets.Let us introduce conjugate momenta for the fields �(x):

�(t, x) = ∂L∂(∂0�(t, x))

(1.18)

(we assume here that � �= 0; there are interesting exceptions, with � = 0, suchas, for example, classical electrodynamics; these have to be discussed separately).Again, the Lorentz and ‘internal space’ indices of the �s and �s are hidden. Thehamiltonian of the system reads

H =∫

d3x H(�,�), H = �∂0�− L (1.19)

and the Poisson bracket of two functionals F1 and F2 of the fields � and � isdefined as

{F1(t, x), F2(t, y)} =∫

d3z

[∂F1(t, x)∂�(t, z)

∂F2(t, y)∂�(t, z)

− ∂F1(t, x)∂�(t, z)

∂F2(t, y)∂�(t, z)

](1.20)

We get, in particular,

{�(t, x),�(t, y)} = −δ(x − y)

{�(t, x),�(t, y)} = {�(t, x),�(t, y)} = 0

}(1.21)

The charge (1.17) can be written as

Qa(t) =∫

d3x �(t, x)(−iT a)�(t, x) (1.22)


and for its Poisson bracket with the field � we get:{Qa(t),�(t, x)

} = (iT a)�(t, x) (1.23)

i.e. indeed the transformation (1.10). Thus, generators of symmetry transforma-tions are conserved charges and vice versa.

Space-time transformations

Finally, we consider transformations (changes of reference frame) which actsimultaneously on the coordinates and the fields. The well-known examplesare, for instance, translations, spatial rotations and Lorentz boosts. In this caseour formalism has to be slightly generalized. We consider an infinitesimaltransformation

x ′µ = xµ + δxµ = xµ + εµ(x) (1.24)

and suppose that the fields �(x) seen in the first frame transform under somerepresentation T (ε) of the transformation (1.24) (which acts on their Lorentzstructure) into �′(x ′) in the transformed frame:

�′(x ′) = exp[−iT (ε)]�(x) (1.25)

(the Lorentz indices are implicit and �′α(x

′) is understood as the αth component ofthe field �(x ′) in the new reference frame). For infinitesimal transformations wehave (∂ ′ denotes differentiation with respect to x ′):

�′(x ′) = �′(x)+ δxµ∂µ�(x)+O(ε2)

∂ ′µ�′(x ′) = ∂µ�

′(x)+ δxν∂µ∂ν�(x)+O(ε2)

}(1.26)

The lagrangian density L′ after the transformation is defined by the equation (see,for example, Trautman (1962, 1996))

S′ =∫�′

d4x ′ L′ (�′(x ′), ∂ ′µ�′(x ′)

)=∫�

d4x[L(�(x), ∂µ�(x)

)− ∂µδ�µ (�(x))

](1.27)

(�′ denotes the image of � under the transformation (1.24) and δ�µ is an arbitraryfunction of the fields) which is a sufficient condition for the new equations ofmotion to be equivalent to the old ones. By this we mean that a change ofthe reference frame has no implications for the motion of the system, i.e. atransformed solution to the original equations remains a solution to the newequations (since (1.27) implies δS′ = δS). In general det(∂x ′/∂x) �= 1 and alsoL′(�′(x), ∂ ′µ�

′(x ′)) �= L

(�(x), ∂µ�(x)

). Moreover, there is an arbitrariness in


the choice of L′ due to the presence in (1.27) of the total derivative of an arbitraryfunction δ�µ(�(x)).

Symmetry transformations are again defined as transformations which leaveequations of motion form-invariant. A sufficient condition is that, for a certainchoice of δ�, the L′ defined by (1.27) satisfies the equation:

L′(�′(x ′), ∂ ′µ�′(x ′)) = L(�′(x ′), ∂ ′µ�

′(x ′)) (1.28)

or, equivalently,

L(�′(x ′), ∂ ′µ�′(x ′)) d4x ′ = [

L(�(x), ∂µ�(x))− ∂µδ�µ(�(x))

]d4x (1.28a)

Note that, if det(∂x ′/∂x) = 1, we recover condition (1.14) up to the total derivative.The most famous example of symmetry transformations up to a non-vanishing totalderivative is supersymmetry (see Chapter 15).

The change of the action under symmetry transformations can be calculated interms of

δL = L(�′(x ′), ∂ ′µ�′(x ′))− L(�(x), ∂µ�(x)) (1.29)

where x and x ′ are connected by the transformation (1.24). Using (1.26) we get

δL = ∂L∂�

δ0�+ ∂L∂(∂µ�)

δ0(∂µ�)+ δxµ∂µL+O(ε2) (1.30)

where

δ0�(x) = �′(x)−�(x) (1.31)

and, therefore, since δ0 is a functional change,

δ0∂µ� = ∂µδ0� (1.32)

Eq. (1.30) can be rewritten in the following form:

δL = δxµ∂µL+[∂L∂�

− ∂µ∂L

∂(∂µ�)

]δ0�+ ∂µ

(∂L

∂(∂µ�)δ0�

)(1.33)

Since

det

(∂x ′µ∂xν

)= 1 + ∂µδxµ (1.34)

we finally get (using (1.28a), (1.33) and the equations of motion):

0 = L∂µδxµ + δxµ∂µL+ ∂µ

(∂L

∂(∂µ�)δ0�

)+ ∂µδ�

µ +O(ε2)

= Lδxµ + ∂L∂(∂µ�)

δ0�+ δ�µ +O(ε2) (1.35)


Re-expressing δ0� in terms of δ� = �′(x ′)−�(x) = δ0�+δxµ∂µ� we concludethat

∂µ jµ ≡ ∂µ

{[Lgµ

ρ −∂L

∂(∂µ�)∂ρ�

]δxρ + ∂L

∂(∂µ�)δ�+ δ�µ

}= 0 (1.36)

This is a generalization of the result (1.16) to space-time symmetries and bothresults are the content of the Noether’s theorem which states that symmetries of aphysical system imply conservation laws.

Examples

We now consider several examples. Invariance of a physical system undertranslations

x ′µ = xµ + εµ (1.37)

implies the conservation law

∂µ�µν(x) = 0 (1.38)

where �µν is the energy–momentum tensor

�µν(x) = ∂L∂(∂µ�)

∂ν�− gµνL (1.39)

(we have assumed that the considered system is such that L′(x ′) defined by (1.27)with � ≡ 0 satisfies L′(x ′) = L(x ′); note also that det(∂x ′/∂x) = 1). The fourconstants of motion

Pν =∫

d3x �0ν(x) (1.40)

are the total energy of the system (ν = 0) and its momentum vector (ν = 1, 2, 3).It is easy to check that transformations of the field �(x) under translations in timeand space are given by

{Pµ,�(x)} = − ∂

∂xµ

�(x) (1.41)

so that �′(x) = �(x)+ {Pµ,�(x)} εµ.The energy–momentum tensor defined by (1.39) is the so-called canonical

energy–momentum tensor. It is important to notice that the physical interpretationof the energy–momentum tensor remains unchanged under the redefinition

�µν(x) → �µν(x)+ f µν(x) (1.42)

where f µν(x) are arbitrary functions satisfying the conservation law

∂µ f µν(x) = 0 (1.43)


with vanishing charges ∫d3x f 0ν(x) = 0 (1.44)

Then (1.38) and (1.40) remain unchanged. A solution to these constraints is

f µν(x) = ∂ρ f µνρ (1.45)

where f µνρ(x) is antisymmetric in indices (µρ). This freedom can be used to makethe energy–momentum tensor symmetric and gauge-invariant (see Problem 1.3).

An infinitesimal Lorentz transformation reads

x ′µ ≡ �µν(ω)xν ≈ xµ + ωµ

νxν (1.46)

(with ωµν antisymmetric) and it is accompanied by the transformation on the fields

�′(x ′) = exp(

12ωµν�

µν)�(x) ≈ (

1 + 12ωµν�

µν)�(x) (1.47)

where �µν is a spin matrix. For Lorentz scalars � = 0, for Dirac spinors �µν =(−i/2)σµν = 1

4 [γ µ, γ ν] (for the notation see Appendix A). Inserting (1.46) and(1.47) into (1.36) (using (1.38) and replacing the canonical energy–momentumtensor by a symmetric one and finally using the antisymmetry of ωµν), we get thefollowing conserved currents:

for a scalar field:

Mµ,νρ(x) = xν�µρ − xρ�µν (1.48)

for a field with a non-zero spin:

Mµ,νρ(x) = xν�µρ − xρ�µν + ∂L∂(∂µ�)

�νρ� (1.49)

The conserved charges

Mνρ(t) =∫

d3x M0,νρ(t, x) (1.50)

are identified with the total angular momentum tensor of the considered physicalsystem. The charges Mµν are the generators of Lorentz transformations on thefields �(x) with

{Mµν,�(x)} = −(xµ∂ν − xν∂µ +�µν)�(x) (1.51)

so that �′(x) = �(x)− 12ωµν {Mµν,�(x)}.

The tensor Mµν is not translationally invariant. The total angular momentumthree-vector is obtained by constructing the Pauli–Lubanski vector

Wµ = 12εµνρκ Mνρ Pκ (1.52)


(where εµνρκ is the totally antisymmetric tensor defined by ε0123 = −ε0123 = 1)which reduces in the rest frame Pµ = (m, 0) to the three-dimensional total angularmomentum Mk ≡ 1

2εi jk Mi j : Mk = W k/m.

1.2 Classical field equations

Scalar field theory and spontaneous breaking of global symmetries

In this section we illustrate our general considerations with the simplest example:the classical theory of scalar fields �(x). We take these to be complex, inorder to be able to discuss a theory which is invariant under continuous globalsymmetry of phase transformations (U (1) symmetry group of one-parameter,unitary, unimodular transformations):

�′(x) = exp(−iq�)�(x) ≈ (1 − iq�)�(x) (1.53)

Thus, the field �(x) (its complex conjugate ��(x)) carries the internal quantumnumber q (−q). It is sometimes also useful to introduce two real fields

φ = 1√2(�+��), χ = −i√

2(�−��) (1.54)

The lagrangian density which satisfies all the constraints of the previous sectionreads:

L = ∂µ�∂µ�� − m2�� − 1

2λ(��)2 − 1

3m21

(��)3 + · · · = T − V (1.55)

where T is the kinetic energy density and V is the potential energy density. Theterms which are higher than second powers of the bilinear combination ��

require on dimensional grounds couplings with dimension of negative powers ofmass.†

Using the formalism of the previous section we derive the equation of motion(since we have two physical degrees of freedom, the variations in � and �� areindependent):

(�+ m2)�+ λ��+ 1

m21

(��)2�+ · · · = 0 (1.56)

In the limit of vanishing couplings (free fields) we get the Klein–Gordon equation.The conserved U (1) current

jµ(x) = iq(��∂µ�−�∂µ��) (1.57)

† In units c = 1, the action has the dimension [S] = g · cm = [h]. Thus [L] = g · cm−3, [�] = (g/cm)1/2 andthe coefficients [m2] = cm−2, [λ] = (g · cm)−1, [m2

1] = g2. In units c = h = 1 we have [L] = g4, [�] = g,

[m2] = [m21] = g2 and S and λ are dimensionless.

1.2 Classical field equations 21

and the canonical energy–momentum tensor

�µν = ∂µ�∂ν�� + ∂ν�∂µ�

� − gµν(∂ρ�∂ρ��)

+ gµνm2�� + 12λgµν(��)2 + · · · (1.58)

are obtained from (1.16) and (1.33), respectively. This tensor can be written inother forms, for example, by adding the term ∂ρ(gµν�∂ρ�� − gρν�∂µ��) andusing the equation of motion. Invariance under Lorentz transformation gives theconserved current (1.48).

The U (1) symmetry of our scalar field theory can be realized in the so-calledWigner mode or in the Goldstone–Nambu mode, i.e. it can be spontaneouslybroken. The phenomenon of spontaneous breaking of global symmetries is wellknown in condensed matter physics. The standard example of a physical theorywith spontaneous symmetry breakdown is the Heisenberg ferromagnet, an infinitearray of spin 1

2 magnetic dipoles. The spin–spin interactions between neighbouringdipoles cause them to align. The hamiltonian is rotationally invariant but theground state is not: it is a state in which all the dipoles are aligned in somearbitrary direction. So for an infinite ferromagnet there is an infinite degeneracy ofthe vacuum.

We say that the symmetry G is spontaneously broken if the ground state (usuallycalled the vacuum) of a physical system with symmetry G (i.e. described by alagrangian which is symmetric under G) is not invariant under the transformationsG. Intuitively speaking, the vacuum is filled with scalars (with zero four-momenta)carrying the quantum number (s) of the broken symmetry. Since we do not wantspontaneously to break the Lorentz invariance, spontaneous symmetry breakingcan be realized only in the presence of scalar fields (fundamental or composite).

Let us return to the question of spontaneous breaking of the U (1) symmetry inthe theory defined by the lagrangian (1.55). The minimum of the potential occursfor the classical field configurations such that

∂V

∂�= ∂V

∂��= 0 (1.59)

For m2 > 0 the minimum of the potential exists for � = �0 = 0 and V (�0) = 0.However, for m2 < 0 (and m2

1 > 0; study the potential for m21 < 0!) the solutions

to these equations are (note that λ must be positive for the potential to be boundedfrom below)

�0(�) = exp(i�)

(−m2

λ

)1/2 [1 +O

(m2

λ2m21

)](1.60)

with V (�0) = −(m4/2λ). The dependence of the potential on the field � is shownin Fig. 1.1. Thus, the U (1) symmetric state is no longer the ground state of this


Fig. 1.1.

theory. Instead, the potential has its minimum for an infinite, degenerate set ofstates (one often calls it a ‘flat direction’) connected to each other by the U (1)rotations. In this example we can choose any one of these vacua as the groundstate of our theory with spontaneously broken U (1) symmetry.

Physical interpretation of this model (and of the negative mass parameter m2)is obtained if we expand the lagrangian (1.55) around the true ground state,i.e. rewrite it in terms of the fields (we choose � = 0 and put m2

1 = ∞):�(x) = �(x) − �0. After a short calculation we see that the two dynamicaldegrees of freedom, φ(x) and χ(x) (see (1.54)), have the mass parameters (−m2)

(i.e. positive) and zero, respectively (see Chapter 11 for more details). Theχ(x) describes the so-called Goldstone boson – a massless mode whose presenceis related to the spontaneous breaking of a continuous global symmetry. Thedegenerate set of vacua (1.60) requires massless excitations to transform onevacuum into another one. We shall return to this subject later (Chapter 9).

In theories with spontaneously broken symmetries the currents remain conservedbut the charges corresponding to the broken generators of the symmetry group areonly formal in the sense that, generically, the integral (1.17) taken over all spacedoes not exist. Still, the Poisson brackets (or the commutators) of a charge witha field can be defined provided we integrate over the space after performing theoperation at the level of the current (see, for example, Bernstein (1974)).

Spinor fields

The basic objects which describe spin 12 particles are two-component anticom-

muting Weyl spinors λα and χ α, transforming, respectively, as representations( 1

2 , 0) and (0, 12) of the Lorentz group or, in other words, as two inequivalent

representations of the SL(2,C) group of complex two-dimensional matrices M


of determinant 1:

λ′α(x′) = M β

α λβ(x) (1.61)

χ ′α(x ′) = (M†−1

)αβχ β (x) (1.62)

where x ′ and x are connected by the transformation (1.46),

M = exp[−(i/4)ωµνσ

µν]

(1.63)

and the two-dimensional matrices σµν are defined in Appendix A. We use thesame symbol for two- and four-dimensional matrices σµν since it should not leadto any confusion in every concrete context. Two other possible representationsof the SL(2,C) group denoted as λα and χα and transforming through matrices(M−1)T and M∗ are unitarily equivalent to the representations (1.61) and (1.62),respectively. The corresponding unitary transformations are given by:

λα = εαβλβ, χ α = χβεβα (1.64)

where the antisymmetric tensors εαβ and εαβ are defined in Appendix A. Sincetransition from the representation (1.61) to (1.62) involves complex conjugation itis natural to denote one of them with a bar. The distinction between the undottedand dotted indices is made to stress that the two representations are inequivalent.The pairwise equivalent representations correspond to lower and upper indices.The choice of (1.61) and (1.62) as fundamental representations is dictated by ourconvention which identifies (A.32) with the matrix M in (A.16). Those unfamiliarwith two-component spinors and their properties should read Appendix A beforestarting this subsection.

Scalars of SL(2,C) can now be constructed as antisymmetric products ( 12 , 0)⊗

( 12 , 0) and (0, 1

2)⊗(0, 12). They have, for example, the structure εαβλβφα = λαφα =

−λαφα ≡ λ · φ or χ β ηαεβα = χαη

α = −χ αηα ≡ χ · η. Thus, the tensors εαβ and

εαβ are metric tensors for the spinor representations. A four-vector is a product of( 1

2 , 0) and (0, 12) representations, for example, εαβλβ(σ

µ)αβ χβ = λα (σµ)αβ χ β or

χα (σ µ)αβ λβ (see Appendix A for the definitions of σµ and σ µ).We suppose now that a physical system can be described by a Weyl field (or a set

of Weyl fields) λα which transforms as ( 12 , 0) under the Lorentz group. Its complex

conjugates are λα ≡ (λα)� (see the text above (1.64)) and λα = λβε

βα which

transforms as (0, 12) representation. (Note, that numerically

{εαβ

} = {εαβ} = iσ 2.)It is clear that SL(2,C)-invariant kinetic terms read

L = iλσµ∂µλ = iλσ µ∂µλ (1.65)

where the two terms are equal up to a total derivative. The factor i has been


introduced for hermiticity of the lagrangian since one assumes that the spinorcomponents are anticommuting c-numbers. Indeed, this is what we have toassume for these classical fields when we follow the path integral approach to fieldquantization (see Chapter 2).

Furthermore, if the field λ carries no internal charges (or transforms as a realrepresentation of an internal symmetry group), one can construct the Lorentzscalars λλ = λαλα and λλ = λαλ

α and introduce the so-called Majorana massterm

Lm = − 12 m

(λλ+ λλ

)(1.66)

The most general hermitean combination, − 12 m1(λλ + λλ) − (i/2)m2(λλ − λλ),

can be recast in the above form with m = (m21 + m2

2)1/2 by the transformation

λ → exp(iϕ)λ with exp(2iϕ) = (m1 − im2)/(m21 + m2

2)1/2. The term Lm does not

vanish since λαs are anticommuting c-numbers. Finally, we remark that a systemdescribed by a single Weyl spinor has two degrees of freedom, with the equationof motion obtained from the principle of minimal action (variations with respect toλ and λ are independent): (

iσ µ∂µλ)α= mλα (1.67)

We now consider the case of a system described by a set of Weyl spinors λiα

which transform as a complex representation R (see Appendix E for the definitionof complex and real representations) of a group of internal symmetry. Such theoryis called chiral. Then, the kinetic lagrangian remains as in (1.65) but no invariantmass term can be constructed since Lorentz invariants λiλ j and λi λ j are notinvariant under the symmetry group. Note that in this case the spinors λα (and λα),i.e. the spinors transforming as (0, 1

2) (or its unitary equivalent) under the Lorentztransformations, form the representation R�, i.e. complex conjugate to R.

There are also interesting physical systems that consist of pairs of independentsets of Weyl fields λα and λc

α (the internal symmetry indices i , for example, λiα, and

the summation over them are always implicit), both being ( 12 , 0) representations

of the Lorentz group but with λcα transforming as the representation R�, complex

conjugate to R. Such theory is called vector-like. We call such spinors chargeconjugate to each other. The operation of charge conjugation will be discussedin more detail in Section 1.5. The mass term can then be introduced and the fulllagrangian of such a system then reads:

L = iλσ µ∂µλ+ iλcσµ∂µλc − m

(λλc + λλc

)(1.68)

Similarly to the case of Majorana mass, the mass term −im ′(λλc− λλc), if present,can be rotated away by the appropriate phase transformation. Remember thatλcσµ∂µλ

c = λcσ µ∂µλc (up to a total derivative), so both fields λ and λc enter the


lagrangian in a fully symmetric way. The mass term, the so-called Dirac mass, isnow invariant under the internal symmetry group. The equations of motion whichfollow from (1.68) are: (

iσ µ∂µλ)α= mλc

α (1.69)

and similarly for λc. Thus, the mass term mixes positive energy solutions for λ withnegative ones for λc (which are represented by positive energy solutions for λc

α) andvice versa. In accord with the discussion of Appendix A, this can be interpreted asa mixing between different helicity states of the same massless particle.

As we shall discuss in more detail later, the electroweak theory is chiraland quantum chromodynamics is vector-like. In both cases, it is convenient totake as the fundamental fields of the theory Weyl spinors which transform as( 1

2 , 0) representations of SL(2,C). We recall that (as discussed in Appendix A)positive energy classical solutions for spinors λi

αs describe massless fermionstates with helicity − 1

2 and, therefore, λiαs are called left-handed chiral fields.

In the electroweak theory (see Chapter 12) all the left-handed physical fieldsare included in the set of λi

αs which transforms then as a reducible, complexrepresentation R of the symmetry group G = SU (2) × U (1) (R �= R�). To bemore specific, in this case the set λi

α includes pairs of left-handed particles withopposite electric charges such as, for example, e−, e+ and quarks with electriccharges ∓ 1

3 , ± 23 , i.e. this set forms a real representation of the electromagnetic

U (1) gauge group, but transforms as a complex representation R under the fullSU (2) × U (1) group. Negative energy classical solutions for λi

αs, in the Diracsea interpretation, describe massless helicity + 1

2 states transforming as R�. Eachpair of states described by λi

α is connected by the C P transformation (see later)and can therefore be termed particle and antiparticle. These pairs consist of thestates with opposite helicities. (Whether within a given reducible representationcontained in R the − 1

2 or + 12 helicity states are called particles is, of course, a

matter of convention.)An equivalent description of the same theory in terms of the fields λi α transform-

ing as (0, 12) representations of SL(2,C) and as R� under the internal symmetry

group can be given. In this formulation positive (negative) energy solutions for λi αsdescribe all + 1

2 (− 12 ) helicity massless states of the theory (formerly described as

negative (positive) energy solutions for λiαs).

In the second case (QCD) the fundamental fields of the theory, the left-handedλiαs and λci

α s, transform as 3 and 3� of SU (3), respectively, and are related by chargeconjugation. Positive energy solutions for λi

αs are identified with − 12 helicity states

of massless quarks and positive energy solutions for λciα s describe − 1

2 helicitystates of massless antiquarks. Negative energy solutions for λi

αs and λciα s describe,


respectively, antiquarks and quarks with helicity + 12 . Since the representation R is

real, mass terms are possible.For a vector-like theory, like QCD, with charge conjugate pairs of Weyl spinors

λ and λc, both transforming as ( 12 , 0) of the Lorentz group but as R and R�

representations of an internal symmetry group, it is convenient to use the Diracfour-component spinors (in the presence of the mass term in (1.68) λ and λc can beinterpreted as describing opposite helicity (chirality) states of the same masslessparticle which mix with each other due to the mass term)

� =(

λα

λcα

)or �c =

(λcα

λα

)(1.70)

transforming as R and R�, respectively, under the group of internal symmetries.The lagrangian (1.68) can be rewritten as

L = i�γ µ∂µ� − m�� (1.71)

where we have introduced � ≡ (λcα, λα) to recover the mass term of (1.68).Writing � = �†γ 0 and comparing the kinetic terms of (1.68) and (1.71) we definethe matrices γ µ. They are called Dirac matrices in the chiral representation and aregiven in Appendix A. Also in Appendix A we derive the Lorentz transformationsfor the Dirac spinors. Another possible hermitean mass term, −im�γ5�, can berotated away by the appropriate phase transformation of the spinors λ and λc.

Four-component notation can also be introduced and, in fact, is quite convenientfor Weyl spinors λα and χ α transforming as ( 1

2 , 0) and (0, 12) under the Lorentz

group. We define four-component chiral fermion fields as follows

�L =(λα

0

)�R =

(0χ α

)(1.72)

They satisfy the equations

�L = 1 − γ5

2�L ≡ PL�L, PR�L = 0

�R = 1 + γ5

2�R ≡ PR�R, PL�R = 0

(1.73)

which define the matrix γ5 in the chiral representation. By analogy with thenomenclature introduced for λα and χ α the fields �L and �R are called left-and right-handed chiral fields, respectively. Possible Lorentz invariants are of theform �R�L ≡ χαλα and �L�R ≡ λαχ

α. The invariants λαλα and λαλα can

be written as �TL C�L and �LC�T

L , respectively, with the matrix C defined in(A.104). In particular, for a set of Weyl spinors λα and their complex conjugate


λα, transforming as R and R�, respectively, under the internal symmetry group, thesets†

� iL =

(λiα

0

)�ci

R =(

0λi α

)(1.74)

also transform as R and R�, respectively. For instance, in ordinary QED if �L

has a charge −1 (+1) under the U (1) symmetry group, we can identify it with theleft-handed chiral electron (positron) field. The �c

R is then the chiral right-handedpositron (electron) field.

Lorentz invariants � iL�

cjR and �ci

R �j

L (or a combination of them) are genericallynot invariant under the internal symmetry group but such invariants can beconstructed by coupling them, for example, to scalar fields transforming properlyunder this group (see later). The kinetic part of the lagrangian for the chiral fieldsis the same as in (1.71) but no mass term is allowed.

We note that γ 5 = γ5 = iγ 0γ 1γ 2γ 3 and that the following algebra is satisfied:

{γ µ, γ ν} = 2gµν,{γ 5, γ µ

} = 0 (1.75)

Finally, for objects represented by a Weyl field λα with no internal chargesor transforming as a real representation of an internal symmetry group we canintroduce the so-called Majorana spinor

�M =(λα

λα

)(1.76)

In this notation the lagrangian given by (1.65) and (1.66) reads

LM = i

2�Mγ µ∂µ�M − m

2�M�M (1.77)

The equations of motion for free four-dimensional spinors have the form(iγ µ∂µ − m

)� = 0 (1.78)

for the Dirac spinors and with m = 0 for the chiral spinors. In the latter case it issupplemented by the conditions (1.73). The solutions to the equations of motionfor the Weyl spinors and for the four-component spinors in the chiral and Diracrepresentations for the γ µ matrices are given in Appendix A.

† The notation in (1.74) has been introduced by analogy with (1.70). Note, however, that the superscript ‘c’in the spinors �ci

R is now used even though λα and λα are not charge conjugate to each other (since theyare in different Lorentz group representations). Thus, unlike the spinors in (1.70) the spinors in (1.74) aredefined for both chiral and vector-like spectra of the fundamental fields. For a vector-like spectrum we have�ci

R = (�ci )R.


A few more remarks on the free Dirac spinors should be added here. Themomentum conjugate to � is

�α = ∂L∂(∂0�α)

= i�†α (1.79)

For the hamiltonian density we get

H = �∂0� − L = �†γ 0(−iγγγ · ∂∂∂ + m)� (1.80)

and the energy–momentum tensor density reads

�µν = i�γ µ ∂

∂xν

� (1.81)

((1.81) is obtained after using the equations of motion). The angular momentumdensity

Mκ,µν = i�γ κ

(xµ ∂

∂xν

− xν ∂

∂xµ

− i

2σµν

)� (1.82)

(where four-dimensional σµν = (i/2)[γµ, γν]) gives the conserved angular mo-mentum tensor

Mµν =∫

d3x M0µν (1.83)

The lagrangian (1.65) can be extended to include various interaction terms. Forinstance, for a neutral (with respect to the internal quantum numbers) Weyl spinorλ interacting with a complex scalar field φ (also neutral) the dimension four ([m]4

in units h = c = 1) Yukawa interaction terms are:

λλφ, λλφ�, λλφ�, λλφ (1.84)

Similar terms can be written down for several spinor fields (λ, χ, . . .) interactingwith scalar fields. Additional symmetries, if assumed, would further constrain theallowed terms. These may be internal symmetries or, for instance, supersymmetry(see Chapter 15). The simplest supersymmetric theory, the so-called Wess–Zuminomodel, describing interactions of one complex scalar field φ(x) with one chiralfermion λ(x) has the following lagrangian†

LWZ = iλσ · ∂λ− 12 m(λλ+ λλ)+ ∂µφ

�∂µφ − m2φ�φ

− g(φλλ+ φ�λλ)− gmφ�φ(φ� + φ)− g2(φ�φ)2 (1.85)

The absence of otherwise perfectly allowed coupling −g′(φ�λλ + φλλ) is dueto supersymmetry. In the four-component notation we get (φλλ + φ�λλ) =�M PL�Mφ + �M PR�Mφ�, where �M is given by (1.76).

† This form of the lagrangian follows after eliminating auxiliary fields via their (algebraic) equations of motion,see Chapter 15.

1.3 Gauge field theories 29

In general, in the four-component notation the Yukawa terms have the genericstructure (� i

R�j

L ± �j

L�iR)�

k(�k�) where the chiral fields are defined by (1.72)and the internal symmetry indices are properly contracted.

If λi s transform as a representation R (with hermitean generators T a) of aninternal symmetry group, the interactions with vector fields Aa

µ in the adjointrepresentation of the group (as for the gauge fields, see Section 1.3) take the form:

λi σ µ(T a)j

i λ j Aaµ = � i

Lγµ(T a)

ji �L j Aa

µ

with �L given by (1.72).For a vector-like theory, the lagrangian (1.68) in which λci transforms as a

representation R∗ of the group (with generators −T a∗, see Appendix E) can besupplemented by terms like:

λi σ µ(T a)j

i λ j Aaµ + λc

i σµ(−T a∗)i

jλc j Aa

µ

= λi σ µ(T a)j

i λ j Aaµ + λc jσµ(T a) i

j λcj Aa

µ

= � iγ µ(T a)j

i � j Aaµ

In the last form (in which �(x) is given by (1.70)) this interaction can beadded to the lagrangian (1.71). For the abelian field Aµ this is the same as theminimal coupling of the electron to the electromagnetic field (pµ → pµ + eAµ).Other interaction terms of a Dirac field (1.70), for example, with a scalar field�(x) or a vector field Aµ(x), are possible as well. A Lorentz-invariant andhermitean interaction part of the lagrangian density may consist, for example, ofthe operators:

��, �γ 5��, �γ 5γµ Aµ� (1.86)

1

M��,

1

M�γµ Aµ��,

1

M��Aµ Aµ,

1

M2�� etc.

(1.87)The first two terms are Yukawa couplings (scalar and pseudoscalar). The terms in(1.87) require dimensionful coupling constants, with the mass scale denoted by M .The last term, describing the self-interactions of a Dirac field, is called the Fermiinteraction. It is straightforward to include the interaction terms in the equation ofmotion.

1.3 Gauge field theories

U (1) gauge symmetry

As a well-known example, we consider gauge invariance in electrodynamics. It isoften introduced as follows: consider a free-field theory of n Dirac particles with


the lagrangian density

L =n∑

i=1

(�i iγ

µ∂µ�i − m�i�i)

(1.88)

Then define a U (1) group of transformations on the fields by

� ′i (x) = exp(−iqi�)�i (x) (1.89)

where the parameter qi is an eigenvalue of the generator Q of U (1) and numbersthe representation to which the field �i belongs. The lagrangian (1.88) is invariantunder that group of transformations.

By Noether’s theorem (see Section 1.1), the U (1) symmetry of the la-grangian (1.88) implies the existence of the conserved current

jµ(x) =∑

i

qi�iγµ�i (1.90)

and therefore conservation of the corresponding charge (1.17). We now considergauge transformations (local phase transformations in which � is allowed to varywith x)

� ′i (x) = exp[−iqi�(x)]�i (x) (1.91)

It is straightforward to verify that lagrangian (1.88) is not invariant under gaugetransformations because transformation of the derivatives of fields gives extra termsproportional to ∂µ�(x). To make the lagrangian invariant one must introduce anew term which can compensate for the extra terms. Equivalently, one should finda modified derivative Dµ�i (x) which transforms like �i (x)

[Dµ�i (x)]′ = exp[−iqi�(x)]Dµ�i (x) (1.92)

and replace ∂µ by Dµ in the lagrangian (1.88).The derivative Dµ is called a covariant derivative. The covariant derivative is

constructed by introducing a vector (gauge) field Aµ(x) and defining†

Dµ�i (x) = [∂µ + iqi eAµ(x)]�i (x) (1.93)

where e is an arbitrary positive constant. The transformation rule (1.92) is ensuredif the gauge field Aµ(x) transforms as:

A′µ(x) = Aµ(x)+ 1

e∂µ�(x) (1.94)

Covariant derivatives play an important role in gauge theories. In particular

† The sign convention in the covariant derivative is consistent with the standard minimal coupling in the classicallimit (Bjorken and Drell (1964); note that e < 0 in this reference).


one may construct new covariant objects by repeated application of covariantderivatives. For the antisymmetric product of two derivatives

[Dµ, Dν]�i = Dµ(Dν�i )− Dν(Dµ�i )

one gets

[Dµ, Dν]�i = iqi e[∂µ Aν(x)− ∂ν Aµ(x)]�i (1.95)

By comparing the gauge transformation properties of both sides in (1.95) weconclude that

Fµν(x) = ∂µ Aν(x)− ∂ν Aµ(x) (1.96)

is gauge-invariant. The field strength tensor Fµν can be used to complete thelagrangian with the gauge-invariant kinetic energy term for the gauge field itself(assume one fermion field for definiteness)

L = �(iD/ − m)� − 14 F2

µν (1.97)

The Euler–Lagrange equations of motion are now

∂µFµν = eq�γ ν�

(i/∂ − m)� = eq A/�

or

(iD/ − m)� = 0

and identifying e (e > 0) with electric charge we recognize in our theory theMaxwell–Dirac electrodynamics.

Of course, U (1) gauge invariance implies global U (1) invariance and theconservation of current (1.90) and the corresponding charge (1.17). It also impliesabsence of the gauge field mass term m2 Aµ Aµ. But such a term does not break theglobal U (1) symmetry.

Exactly analogous considerations apply to scalar field theories with U (1) gaugesymmetry and to theories with Weyl fermions.

Non-abelian gauge symmetry

To construct a non-abelian gauge field lagrangian we repeat the same steps. Westart with the free-field lagrangian for Dirac fields � which transform according toa representation of some non-abelian Lie group

� ′(x) = exp(−i�αT α)�(x) (1.98)

where the T α are the hermitean matrix representations of the generators of thegroup, appropriate for the fields � and satisfying relations (1.11).


The free-field lagrangian is invariant under global group transformations (1.98).Let us now consider the extension of the group G to a group of local gaugetransformations. Generalizing the U (1) case we seek a covariant derivative suchthat

[Dµ�(x)]′ = exp[−i�α(x)T α]Dµ�(x) = U (x)Dµ�(x) (1.99)

By analogy with (1.93) we expect it to be given by a combination of the normalderivative and a transformation on the fields �

Dµ�(x) = [∂µ + Aµ(x)]�(x) (1.100)

where Aµ is a (antihermitean) element of the Lie algebra†

Aµ(x) = +ig Aαµ(x)T

α (1.101)

Thus we need gauge fields in the number given by the number of generators ofthe group. The constant g is arbitrary. In classical physics, g is a dimensionfulparameter and therefore can always be scaled to one (similar arguments apply to(1.93)). At the quantum level g is relevant since quantum theory contains thePlanck constant h and g is dimensionless in units h = 1. Equivalently, at thequantum level the normalization of the field is fixed by the normalization of thesingle particle state. With the form (1.100) the condition (1.99) is satisfied if thegauge transformation rule for Aµ(x) is as follows:

A′µ(x) = U (x)Aµ(x)U

−1(x)− [∂µU (x)]U−1(x)

= U (x)[Aµ(x)+ ∂µ]U−1(x) (1.102)

or for an infinitesimal transformation

δAµ(x) = A′µ(x)− Aµ(x) = ∂µ�(x)+ [Aµ(x),�(x)] (1.103)

where

�(x) = +i�α(x)T α (1.104)

is an infinitesimal gauge parameter in the matrix notation. The correspondingtransformation for the gauge fields Aα

µ(x) reads

δAαµ(x) = (1/g)∂µ�

α(x)+ cαβγ�β(x)Aγ

µ(x) (1.105)

It can be checked that transformations (1.103) and (1.105) form a group. It isalso seen from (1.105) that under global transformations (∂µ� = 0) the gaugefields transform according to the adjoint representation of the group with (T α)βγ =−icαβγ .

† The sign convention for the non-abelian case is consistent with the convention (1.93); this is convenient forembedding electromagnetism into the electroweak theory.


We similarly can construct new covariant quantities by repeated application ofthe covariant derivative. We get, for instance,

[Dµ, Dν]�(x) = (∂µ Aν(x)− ∂ν Aµ(x)+ [Aµ(x), Aν(x)])�(x) (1.106)

Thus, the antisymmetric tensor (the field strength)

Gµν(x) = ∂µ Aν(x)− ∂ν Aµ(x)+ [Aµ(x), Aν(x)] (1.107)

is a covariant quantity transforming under gauge transformations as follows:

G ′µν(x) = U (x)Gµν(x)U

−1(x) (1.108)

For an infinitesimal transformation we get

δGµν(x) = [Gµν(x),�(x)] (1.109)

It is obvious from (1.107) that the tensor Gµν can be decomposed in terms of groupgenerators

Gµν = igGαµνT α (1.110)

where

Gαµν = ∂µ Aα

ν − ∂ν Aαµ − gcαβγ Aβ

µ Aγν (1.111)

The transformation rule for Gαµν(x) follows from (1.108) and (1.109).

Finally, we can generalize the definition of the covariant derivative to apply it toany Lie algebra element ξ(x) = +iξα(x)T α. By applying the covariant derivativeDρ = ∂ρ + Aρ(x) to the object ξ� and insisting on the Leibnitz rule

Dρ(ξ�) = (Dρξ)� + ξ(Dρ�)

we get

Dµξ = ∂µξ + [Aµ, ξ ] (1.112)

Then the gauge transformation (1.103) can be written simply as

δAµ(x) = Dµ�(x) (1.113)

Moreover, we have

[Dµ, Dν]Gρσ = [Gµν, Gρσ ] (1.113a)

We are ready to write down the gauge-invariant lagrangian for a non-abeliangauge field theory with fermions. It reads

L = + 1

2g2Tr[GµνGµν] + �(iD/ − m)� (1.114)

(Tr[GµνGµν] is gauge-invariant because Tr[U GµνGµνU−1] = Tr[GµνGµν]).


Terms of higher order in fields are not allowed for quantum field theory if it isto be renormalizable (Chapter 4). The term Gµ

µ is zero and the possibility of aparity non-conserving term εµνρδGµνGρδ in the lagrangian will be discussed later.The pure gauge field term must be present in the lagrangian because we need thekinetic energy term to be quadratic in the gauge fields. Its gauge-invariant formthen implies the presence of two gauge self-interaction terms of the order g and g2,respectively

1

2g2Tr[GµνGµν] = − 1

4(∂µ Aαν − ∂ν Aα

µ)2 + gcαβγ Aβ

µ Aγν ∂

µ Aνα

− 14 g2cαβγ cασδ Aβ

µ Aγν Aµσ Aνδ (1.115)

(we recall our normalization Tr[T αT β] = 12δ

αβ).We end this section with a derivation of the Euler–Lagrange equations for gauge

fields. One requires the action to be stationary with respect to small variations ofthe gauge fields

Aµ → Aµ + δAµ (1.116)

and its derivative

δ(∂ν Aµ) = ∂ν(δAµ)

where we formally treat δAµ as a covariant quantity transforming in an adjointrepresentation of the group

δA′µ(x) = U (x)δAµ(x)U

−1(x) (1.117)

The change of the field strength (1.107) under the variation (1.116) can beexpressed as follows:

Gµν → Gµν + Dµ(δAν)− Dν(δAµ) (1.118)

Here we have used definition (1.112) of a covariant derivative acting on a covariantobject δAµ. The variation of the gauge field action is

δ

∫d4x

1

2g2Tr[GµνGµν] = 2

g2

∫d4x Tr[Gµν(DµδAν)] (1.119)

(antisymmetry of Gµν in µ and ν has been used). Using the relation

Dµ Tr[(GµνδAν)] = Tr[(DµGµν)δAν] + Tr[Gµν(DµδAν)] (1.120)

and the invariance of Tr[GµνδAν] under gauge transformations (therefore, onthe l.h.s. of (1.120) we may replace the covariant derivative by the ordinary

1.4 From classical to quantum fields 35

derivative; this remark also helps to check relation (1.120)) we conclude, byintegrating (1.120) and neglecting the surface terms at infinity on the l.h.s., that

δ

∫d4x

1

2g2Tr[GµνGµν] = − 2

g2

∫d4x Tr[(DµGµν)δAν] (1.121)

The remaining term

δ

∫d4x �(iD/ − m)�

can be written as

δ

∫d4x �(iD/ − m)� = 2

∫d4x Tr[ jµδAµ] (1.122)

where the current jµ defined by the variation (1.122) reads

jµ(x) = i jαµ(x)Tα

with

jαµ = �γµT α� (1.123)

Combining (1.121) and (1.122) we get the equation of motion

(1/g2)DµGµν − jν = 0 (1.124)

Applying the covariant derivative to (1.124) one derives the covariant divergenceequation for the current (1.123)

Dµ jµ(x) = 0 (1.125)

Thus, gauge fields can only couple consistently to currents which are covariantlyconserved. We also see from (1.125) that the non-abelian charge associated withthe fermionic current is not a constant of motion; since gauge fields are not neutraltheir contribution must be included to find conserved charges.

1.4 From classical to quantum fields (canonical quantization)

In this book we use the path integral approach to the quantization of fields (seeChapter 2). However, for the sake of completeness and easy reference, in thissection we briefly recall the canonical quantization, procedure (for more detailssee Bjorken & Drell (1964), Itzykson & Zuber (1980)).


Scalar fields

We begin with a scalar field theory and consider only free fields. The canonicalquantization of interacting fields can be performed along similar lines in theso-called interaction picture (we refer the reader to the standard textbooks, forexample, Bjorken & Drell (1965); see also Section 2.7). An arbitrary solution tothe free Klein–Gordon equation (1.56) may be expanded as a Fourier integral overplane-wave solutions:

�(x) =∫

d3k

(2π)32k0

[a(k) exp(−ik · x)+ a†(k) exp(ik · x)

](1.126)

where ∫d3k

(2π)32k0=∫

d4k

(2π)42πδ(k2 − m2)θ(k0) (1.127)

is a Lorentz-invariant measure, k0 = E = +(k2 + m2)1/2 and, for a real field�(x), the a(k) and a†(k) are complex conjugate to each other. In the canonicalquantization, the fields and their conjugate momenta � = ∂0� are promoted tohermitean operators in the Heisenberg picture (in a Hilbert space of state vectors)by postulating for them canonical equal-time commutation relations, i.e. replacingthe Poisson brackets in (1.21) by commutators, { , } → −i[ , ]:

[�i (t, x),� j (t, y)] = −iδi jδ(x − y)

[�i (t, x),� j (t, y)] = [�i (t, x),� j (t, y)] = 0

}(1.128)

(the indices (i, j) are internal symmetry indices).It follows from the postulated relations (1.128) that the (hermitean conjugate to

each other) operators a(k) and a†(k) satisfy the following algebra:[a(k), a†(k′)

] = (2π)32k0δ(k − k′)

[a(k), a(k′)] = [a†(k), a†(k′)

] = 0

}(1.129)

The ground state (the vacuum) is defined by the relations:

a(k)|0〉 = 0, 〈0|0〉 = 1 (1.130)

The energy–momentum operator (1.40) can be rewritten in terms of the operatorsa(k) and a†(k):

Pµ =∫

d3x �0µ(x) = 1

2

∫d3k

(2π)32k0kµ[a†(k)a(k)+ a(k)a†(k)

](1.131)

Since a(k)a†(k) = a†(k)a(k) + (2π)32k0δ(0), we see that the vacuum energy isinfinite. For most physical questions (apart from the problem of the cosmological


constant) the vacuum energy is unobservable. Therefore, we redefine the hamilto-nian as

P0 → P0 − 〈0|(a†(k)a(k)+ a(k)a†(k))|0〉 =∫

d3k

(2π)32k0k0a†(k)a(k)

(1.132)The prescription

12(a

†(k)a(k)+ a(k)a†(k)) → : 12(a

†(k)a(k)+ a(k)a†(k)): ≡ a†(k)a(k) (1.133)

is called ‘normal (or Wick’s) ordering’ and is denoted by a double-dot symbol.Its only effect is to subtract from the original expressions the infinite vacuumenergy. For a product of the fields �(x) = �(+)(x) + �(−)(x) we get, forexample, :��: = �(−)�(−) + 2�(−)�(+) + �(+)�(+), where �(+)(�(−)) arepositive (negative) frequency parts of �(x) given by (1.126). It follows from(1.129) and (1.132) that [

Pµ, a†(k)] = kµa†(k) (1.134)

and, consequently,

Pµa†(k)|0〉 = kµa†(k)|0〉 (1.135)

Thus, the state |k〉 ≡ a†(k)|0〉 is an energy–momentum eigenstate. Hence, thea†(k) and a(k) are the creation and annihilation operators, respectively, for aquantum of four-momentum kµ. The field–particle duality now becomes evident.Also, we get

〈k|�(x)|0〉 = 〈0|a(k)�(x)|0〉 = exp(ik · x) (1.136)

i.e. the field �(x) (its negative frequency part) acting on the vacuum creates aparticle at point x , with continuous momentum spectrum and propagating as aplane wave (with positive energy).† When acting on a particle at the point x , thepositive frequency part of the field �(x) annihilates that state into the vacuum andthe negative frequency part creates two-particle state.

Simultaneous measurement of the field strength � at points x and y is possibleonly if

[�(x),�(y)] = 0 (1.137)

† Note that in quantum mechanics, the wave-function �(x) describes a state in the configuration representation,�(x) = 〈x |�(t)〉, whereas in quantum field theory �(x) |0〉 is a state |x〉 of a particle localized at thespace-time point x , with its momentum representation 〈k|�(x)|0〉 = 〈x |k〉†. Hence, in quantum mechanicsthe positive frequency part of �(x) describes propagation with positive energy but, interpreted as a quantumfield, it is its negative frequency part which creates a particle with positive energy.


By direct calculation we get

[�(x),�(y)] =∫

d4k

(2π)42πδ(k2 − m2)ε(k0) exp[−ik · (x − y)] ≡ i (x − y)

(1.138)where

ε(k0) ={+1−1

fork0 > 0k0 < 0

One checks that is a solution of a free Klein–Gordon equation(�x + m2

) (x − y) = 0

(x − y) = − (y − x)

}(1.139)

Since (y − x) vanishes for t1 − t2 = 0, from its Lorentz invariance we concludethat

(x − y) = 0 for all (x − y)2 < 0 (1.140)

Thus the field strength is simultaneously measurable in two points with a space-like interval, i.e. consistently with the notion of locality and causality (two pointswith space-like intervals cannot be connected in a causal way). It follows fromthe postulated commutation relations (1.128) that the quanta of a scalar field obeyBose–Einstein statistics.

Quantization of a complex scalar field �(x) proceeds in quite an analogous way.In fact, the procedure is exactly the same if we work with the two real fields φ andχ defined in (1.54). In terms of the complex field and its conjugate momenta

�i = ∂0�†i , �

†i = ∂0�i (1.141)

we postulate[�i (t, x),� j (t, y)

] = [�

†i (t, x),�†

j (t, y)] = −iδi jδ(x − y) (1.142)

and the other commutators vanish. Fourier integrals, analogous to (1.126), nowread:

�(x) =∫

d3k

(2π)32k0

[a(k) exp(−ik · x)+ b†(k) exp(ik · x)

]�†(x) =

∫d3k

(2π)32k0

[b(k) exp(−ik · x)+ a†(k) exp(ik · x)

] (1.143)

where a(k) and b(k) are two independent, non-hermitean, operator-valued func-tions. They can easily be expressed in terms of the hermitean annihilation andcreations operators a1(k), a2(k) and a†

1(k), a†2(k) for the quanta of the real fields


φ and χ . From (1.142) there follow the commutation relations for operators a(k)and b(k): [

a(k), a†(k′)] = [

b(k), b†(k′)] = (2π)32k0δ(k − k′) (1.144)

Thus, a(k), b(k) and a†(k), b†(k) are also annihilation and creation operators andit is clear from (1.143) that the field �(x) (�†(x)) creates quanta of type b (a) andannihilates those of type a (b). A transparent interpretation of the quanta a and bis obtained by noting that the classical lagrangian (1.55) is invariant under abelianglobal U (1) symmetry: � → exp(−iq�)�, �† → exp(iq�)�†, i.e. the fields� and �† carry opposite U (1) charges.† The conserved charge Q (1.17) can beexpressed in terms of the operators a(k) and b(k):

Q = iq∫

d3x :[�†(x)∂0�(x)− ∂0�

†(x)�(x)]:

= q∫

d3k

(2π)32k0

[a†(k)a(k)− b†(k)b(k)

] ≡ q(Na − Nb) (1.145)

This equation defines the operators Na and Nb which are the number operators forthe quanta a and b carrying opposite charges:

Qa†(k)|0〉 = qa†(k)|0〉Qb†(k)|0〉 = −qb†(k)|0〉

}(1.146)

Note that the charge of the state created by the field �(x) transforming as�(x) → exp(−iq�)�(x) is Q = −q. Moreover, using (1.145) we get[Q,�(x)] = −q�(x), i.e. (1.23) with { , } replaced by −i[ , ]. We canintroduce the unitary operator U = exp(i�Q) ≈ 1 + i�Q such that Ub†(k)|0〉 =exp(−i�q) b†(k)|0〉 and U�(x)U−1 = exp(−i�q)�(x) (we have used (1.146)).A generalization to non-abelian global symmetries is straightforward. These equa-tions can also be interpreted in the following way. We consider the wave-functionin the configuration space of the one-particle state b†(k)|0〉: φ(x) = 〈x |b†(k)|0〉 =〈0|�†(x)b†(k)|0〉 and in the U (1) rotated basis (or of the rotated physical systemin the same basis) φ′(x) = 〈0|�†(x)Ub†(k)|0〉. Using (1.146) we get φ′(x) =exp(−iq�)φ(x). Therefore, the symmetry transformation on the quantum fieldsis the analogue of the transformation on the classical fields interpreted as thewave-functions of one-particle states.

The Feynman propagator

For the sake of definiteness, we again consider scalar field theory with abelianU (1) symmetry and q = 1. One-particle states a (Q = +1) and b (Q = −1) at

† The value of the charge q relative to the charges of other fields can only be fixed in the presence of interactions.


the space-time point (t, x) are created by the operators

�†(t, x)|0〉 ≡ �†(−)(t, x)|0〉�(t, x)|0〉 ≡ �(−)(t, x)|0〉

}(1.147)

where

�†(−)(t, x) =∫

d3k

(2π)32Ea†(k) exp(ik · x)

�(−)(t, x) =∫

d3k

(2π)32Eb†(k) exp(ik · x)

(1.148)

Let the particle a propagate to t ′ > t : the probability amplitude to find it at (t ′, x′)is given by

θ(t ′ − t)〈0|�(t ′, x′)�†(t, x)|0〉= θ(t ′ − t)〈0|�(+)(t ′, x′)�†(−)(t, x)|0〉 (1.149)

This amplitude can also be interpreted as the creation of the particle a at (t, x) andits reabsorption into the vacuum at (t ′, x′). Another way of increasing the chargeby +1 at (t, x) and lowering it by −1 at (t ′, x′) is to create a quantum of charge −1at (t ′, x′) and to propagate it to (t, x) with t > t ′:

θ(t − t ′)〈0|�†(t, x)�(t ′, x′)|0〉= θ(t − t ′)〈0|�†(+)(t, x)�(−)(t ′, x′)|0〉 (1.150)

The Feynman propagator is defined as superposition of the two amplitudes:

G(2)0 (x ′, x) ≡ θ(t ′ − t)〈0|�(t ′, x′)�†(t, x)|0〉

+ θ(t − t ′)〈0|�†(t, x)�(t ′, x′)|0〉≡ 〈0|T (

�†(t, x)�(t ′, x′)) |0〉

=∫

d3k

(2π)32Ek

[θ(t ′ − t) exp[−ik · (x ′ − x)]

+ θ(t − t ′) exp[ik · (x ′ − x)]]

=∫

d4k

(2π)4

i

k2 − m2 + iεexp[−ik · (x ′ − x)] (1.151)

where T denotes the so-called chronological ordering of the field operators.

Spinor fields

Quantization of a Weyl or Dirac field can be based on the phenomenologicalPauli’s exclusion principle for spin 1

2 particles or on the requirement of locality


(i.e. commutativity at the space-like intervals for observables constructed fromthe Dirac spinors; this is the content of the theorem on the connection betweenspin and statistics). It can be also introduced by postulating supersymmetry (acertain symmetry between bosons and fermions, see Chapter 15). We are alwaysconsistently led to the conclusion that the Weyl and Dirac fields must be quantizedby assigning for them certain anticommutation relations.

Using the classical spinors introduced in Section 1.2 and Appendix A, we canwrite down the general wave expansion for solutions of the free Weyl equations forthe massless left-handed field λα(x)

λα(x) =∫

d3k

(2π)32E

[bL(k)aα(k) exp(−ik · x)+ d†

R(k)bα(k) exp(ik · x)](1.152)

and for its hermitean conjugation λ†α(x) (related to the right-handed field through

λβ = λ†αε

βα):

λ†α(x) =

∫d3k

(2π)32E

[b†

L(k)a�α(k) exp(ik · x)+ dR(k)b�

α(k) exp(−ik · x)](1.153)

where aα(k) and bα(k) are positive and negative energy chiral solutions to a freeWeyl equation iσ µ∂µλ(x) = 0 (normalized as in (A.68)) and bL, dR are expansioncoefficients. As for scalar fields, the classical Weyl fields are now promoted tooperators but instead of relations (1.128), we postulate (the canonical momentumconjugate to λα(x) is �α = iλ†

β(σ 0)βα){

λα(t, x), λ†β(t, y)

} = (σ 0)αβδ(x − y) (1.154)

Other anticommutators vanish. The expansion coefficients become operators withthe following anticommutation relations:{

bL(k), b†L(k

′)} = {

dR(k), d†R(k

′)} = (2π)32k0δ(k − k′) (1.155)

and all other anticommutators vanish. Inserting (1.152) into the analogue of (1.81)for the left-handed Weyl fields,

�µν = iλσ µ∂νλ (1.156)

we find

Pµ =∫

d3k

(2π)32Ekµ[b†

L(k)bL(k)− dR(k)d†R(k)

]=∫

d3k

(2π)32Ekµ[b†

L(k)bL(k)+ d†R(k)dR(k)−

{dR(k), d†

R(k)}]

(1.157)


It follows from (1.157) and from the anticommutation relations (1.155) that, withthe vacuum defined by bL(k)|0〉 = dR(k)|0〉 = 0, its energy is not only infinite (asfor the scalar fields) but also is not positive definite. We redefine the vacuum bysubtracting this infinite c-number. This is accomplished by the normal ordering.The normal ordering for the Weyl fields is defined as:

:λ†αλβ : ≡ λ

†(+)α λ

(+)β + λ

†(−)α λ

(+)β + λ

†(−)α λ

(−)β − λ

(−)β λ

†(+)α (1.158)

where the subscript ‘+’ (‘−’) denotes positive (negative) frequency parts of λ andλ† given by (1.152) and (1.153). In consistency with the anticommutation relations(1.155), there is a minus sign for each interchange of λ and λ† required by thenormal ordering. After the normal ordering we get

Pµ =∫

d3x :iλσ 0∂µλ(x):

=∫

d3k

(2π)32Ekµ[b†

L(k)bL(k)+ d†R(k)dR(k)

](1.159)

i.e. we have redefined the vacuum so that its energy is zero. It follows from therelation (1.159) that operators b†

L(k) and d†R(k) have the interpretation of creation

operators of positive-energy states with four-momentum (E, k) which, however,have opposite charges under the global U (1) phase transformations. Indeed,the lagrangian for a single Weyl field has global U (1) symmetry generated bythe transformations λ′ = exp(−iq�)λ. The conserved current jµ = qλσ µλ

(∂ jµ/∂xµ = 0) and the corresponding charge Q = ∫d3x j0(t, x) can be expressed

in terms of the creation and annihilation operators. After the normal ordering weget

Q = q∫

d3x :λ†σ 0λ: = q∫

d3k

(2π)32E:b†

L(k)bL(k)+ dR(k)d†R(k):

= q∫

d3k

(2π)32E

[N+

L (k)− N−R (k)

](1.160)

where

N+L (k) = b†

L(k)bL(k), N−R (k) = d†

R(k)dR(k)

can be interpreted as the number operators for positive-energy state with U (1)charges +q and −q , respectively. Acting with the field λ(x) on the vacuum andprojecting on the momentum eigenstates d†

R(k)|0〉, we see that λα(x) creates ahelicity h = + 1

2 state with charge −q at the point x , described by a superpositionof the plane wave solutions

α〈x|k, h = + 12〉 = 〈k, h = + 1

2 |λα(x)|0〉†= 〈0|dR(k)λα(x)|0〉† = b�α(k) exp(−ik · x)


with four-momenta kµ and positive energy (note, that b�α(k) ∼ a′α(k) is thesolution of (A.60) with positive energy and positive helicity). The field λ

†α(x)

creates a helicity − 12 state with charge +q at the point x – a state which

is a superposition of the plane wave solutions α〈x|k, h = − 12〉 = 〈k, h =

− 12 |λ†

α(x)|0〉† = aα(k) exp(−ik · x).For the Dirac field the procedure described above leads to the field operators

�(x) =∑

s=1,2

∫d3k

(2π)32E

[b(k, s)u(k, s) exp(−ik · x)

+ d†(k, s)v(k, s) exp(ik · x)]

(1.161)

�†(x) =∑

s=1,2

∫d3k

(2π)32E

[b†(k, s)u(k, s)γ0 exp(ik · x)

+ d(k, s)v(k, s)γ0 exp(−ik · x)]

satisfying anticommutation relations{�α(t, x),�†

β(t, y)} = δαβδ(x − y) (1.162)

(α and β are spinor indices) with the other anticommutators vanishing. Thefunctions u(k, s) and v(k, s) are positive and negative energy solutions for thefour-component Dirac equation; s = 1, 2 numbers the + 1

2 and − 12 spin projections

along the chosen quantization axis (see Appendix A). Similar expansions can bewritten down in terms of the helicity amplitudes. For the operators b(k, s) andd(k, s) we get{

b(k, s), b†(k′, s ′)} = {

d(k, s), d†(k′, s ′)} = (2π)32k0δss′δ(k − k′)

and all other anticommutators vanish. Defining the normal ordering for the Diracfields

:�α�β : = �(+)α �

(+)β + �(−)

α �(+)β + �(−)

α �(−)β −�

(−)β �(+)

α (1.163)

where the superscript ‘+’ (‘−’) denotes positive (negative) frequency parts of �

and � given by (1.161), we get for the momentum operator the expression

Pµ =∫

d3x :�0µ(x): =∑

s=1,2

∫d3k

(2π)32Ekµ[b†(k, s)b(k, s)+ d†(k, s)d(k, s)

]The operators b†(k, s) and d†(k, s) have the interpretation of creation operatorsof positive-energy states with four-momentum (E, k) with opposite charges underthe global U (1) phase transformations � ′ = exp(−iq�)�. The conserved current


jµ = q�γ µ� (∂ jµ/∂xµ = 0) and the corresponding charge Q = ∫d3x j0(t, x)

can be expressed in terms of the creation and annihilation operators

Q = q∫

d3x :�†�: = q∑

s=1,2

∫d3k

(2π)32E:b†(k, s)b(k, s)+ d(k, s)d†(k, s):

= q∑

s=1,2

∫d3k

(2π)32E

[N+(k, s)− N−(k, s)

](1.164)

where

N+(k, s) = b†(k, s)b(k, s), N−(k, s) = d†(k, s)d(k, s)

are interpreted as the number operators for positive-energy states with +q and −qU (1) charges, respectively. As for the scalar fields, one can introduce the unitaryoperator U which realizes the U (1) transformations on the quantum states and onthe quantum Weyl and Dirac fields.

Similarly to the case of Weyl fields, acting with the field operator �†α(x) on the

vacuum and projecting on the momentum eigenstates |b(k, s)〉 = b†(k, s)|0〉, wesee that �†

α(x) creates a state of particle b with the spin s and charge +q at thepoint x , described by a superposition of the plane wave solutions

α〈x|b(k, s)〉 = 〈b(k, s)|�†α(x)|0〉†

= 〈0|b(k, s)�†α(x)|0〉† = uα(k, s) exp(−ik · x)

with four-momenta kµ, spin s and positive energy. The operator �α(x), on theother hand, creates a state of particle d with the wave-function given by

α〈x|d(k, s)〉 = 〈d(k, s)|�α(x)|0〉†= 〈0|d(k, s)�α(x)|0〉† = v�

α(k, s) exp(−ik · x)

For chiral fields in the four-component notation one has

�L,R(x) =∫

d3k

(2π)32E

[bL,R(k)uL,R(k) exp(−ik · x)+d†

R,L(k)vR,L(k) exp(ik · x)]

(1.165)where uL,R (vR,L) are positive (negative) energy solutions to the Dirac equation(1.78), which satisfy the chirality conditions PL,RuL,R = uL,R, PL,RvR,L = vR,L.Obviously, in the chiral representation for the Dirac matrices we have

uL(k) = a(k)

00

, uR(k) = 0

0a′(k)

(1.166)

and similar expressions for vR and vL with a(k) and a′(k) replaced by b(k) andb′(k), respectively. (We recall that a(k), a′(k), b(k), b′(k) are solutions to the Weyl


equations.) Thus, a chiral left-handed massless field �L creates a particle withcharge (−q) and positive helicity and �

†L creates a particle with charge (+q) and

negative helicity (see the remarks below (A.62)).A self-conjugate four-component field, (1.76) (neutral under global symmetry

transformations) has the following expansion

�M(x) =∑

s=1,2

∫d3k

(2π)32E

[b(k, s)u(k, s) exp(−ik · x)

+ exp(iρ)b†(k, s)v(k, s) exp(ik · x)]

(1.167)

�†M(x) =

∑s=1,2

∫d3k

(2π)32E

[b†(k, s)u(k, s)γ0 exp(ik · x)

+ exp(−iρ)b(k, s)v(k, s)γ0 exp(−ik · x)]

exp(iρ) is a phase factor depending on the choice of the phase for one-particlestates (Kayser 1994).

Symmetry transformations for quantum fields

One way to discuss symmetry transformations for quantum fields is to interpretsolutions to the classical field equations as wave-functions of quantum states (seethe remark below (1.146)). The following results apply to free and interactingfields. We begin with the Lorentz transformations and consider two observers intwo different reference frames O and O′, and by analogy between classical andquantum physics we assume the transformation law (1.25) for wave-functions (inconfiguration space) of quantum states. First of all, it is obvious that a given (thesame) physical quantum system is seen by both observers. It is described by themby the state vectors which form two isomorphic to each other but, in general,different Hilbert spaces. We shall make here the following strong assumptionwhich is a necessary (but not sufficient) condition for the change of the referenceframe to be a symmetry transformation: the two Hilbert spaces are identical and canbe identified with each other (in consequence, the operators acting in one of themcan also be identified with suitable operators acting in the other space). Thereforethe Hilbert space used by observer O is mapped onto itself by a unitary operator Uwhich is defined by the equation:

|v′〉 = U |v〉 (1.168)

In the frame O′ the state vector |v′〉 describes the same physical state as doesthe state vector |v〉 in O. The state vectors form a representation of the Lorentzgroup. We note already here that the assumption about the identity of the two


Hilbert spaces is not necessarily satisfied for such physically relevant changes ofthe reference frame as, for example, space reflection (to be discussed later) whichis not a symmetry of many physical systems.

Our goal is to find the action of the operator U on the field operators �(x) whichmay carry a Lorentz index α. For Poincare transformations x ′µ = �µ

νxν ≈ xµ +εµ + ωµ

νxν , (1.47) interpreted as the equation for the one-particle wave-functionsreads:

φ′α(x′) ≡ α〈x ′|v′〉 =

(exp( 1

2ωµν�µν)

) β

α β〈x |v〉 (1.169)

where

α〈x | = 〈0|�α(x)

α〈x ′| = 〈0|�α(x ′)

|v′〉 = U (ε, ω)|v〉

(1.170)

(the index α refers to the original or to the new reference frame; in the first andthe second equation of (1.170) the same field operator � appears since the Hilbertspaces of both observers are identical). Therefore,

U�α(x)U−1 = (

exp(− 12ωµν�

µν)) β

α�β(x

′) (1.171)

(Internal quantum number indices may be implicit.†)For translations, (1.171) simplifies to:

U�α(x)U−1 = �α(x

′) (1.172)

Writing

U (ε) = exp(iεµPµ

)U (ω) = exp

(− i

2ωµν Mµν

) (1.173)

we obtain the infinitesimal form of these transformations

i[Pµ,�(x)] = ∂�(x)

∂xµ(1.174)

and

i[Mµν,�(x)

] = (xµ ∂

∂xν

− xν ∂

∂xµ

+�µν

)�(x) (1.175)

which replace the Poisson brackets (1.41) and (1.51). These equations suggest

† Also, one often defines the operator �′α(x

′) = U−1�α(x)U such that the matrices satisfy the equation

〈v2|�′α(x

′)|v1〉 = (exp( 1

2ωµν�µν)

) βα〈v2|�β(x)|v1〉 or the operator �′

α(x) = U�(x)U−1 such that〈v′2|�′(x)|v′1〉 = 〈v2|�(x)|v1〉.


therefore the identification of the generators Pµ and Mµν with the four-momentumvector and the angular momentum tensor, respectively. In every concrete theorythose operators can be constructed using equations like (1.131) or (1.132) or(1.157) and one can check explicitly if (1.174) and (1.175) follow from thecommutation relations imposed in the quantization procedure. If so, then the theoryis Lorentz-invariant.

An interesting exception to relations (1.174) and (1.175) is the quantization ofthe gauge fields in non-covariant gauges (for example, in the Coulomb gauge).Nevertheless, the unitary operators U (ε, ω) can be found and the theory, althoughnot manifestly so, is Lorentz-invariant. The Lorentz boosts must be supplementedby additional gauge transformations (to recover the non-covariant gauge in the newframe). See, for example, Bjorken & Drell (1965).

The U (1) transformations in the internal quantum number space on the freequantum fields have already been discussed in the first subsection of this section.The results obtained there remain generally valid for non-abelian groups of trans-formations and for interacting fields. We may follow the strategy of interpreting(1.9) as the equation relating the wave-functions of one-particle states in theoriginal and rotated basis built by the action of the field operator �(x) on thevacuum: |i〉 = �i (x)|0〉 (the index i refers to the original or rotated basis). Wehave then:

φi (x) ≡ 〈i |v′〉 = (exp(−i�aT a)

)i j〈 j |v〉 (1.176)

where the states |v〉 and |v′〉 = U |v〉 describe the same physical system before andafter the rotation of the basis (we again assume that the two Hilbert spaces can beidentified with each other). Therefore, using (1.176), we find the transformationlaw for the operators

U�i (x)U−1 = (

exp(−i�aT a))

i j� j (x) (1.177)

In deriving the last equation we have assumed that the vacuum is invariant underthe considered group of transformations: U |0〉 = |0〉. Later we shall also discusstheories with spontaneously broken global symmetries (see Chapter 9) where theground state of the system (the physical vacuum determined by the equations ofmotion) is not invariant under the symmetry group of the lagrangian. Eq. (1.177)then remains valid for the original fields of the lagrangian but not for the physicalfields defined as perturbation around the true vacuum.

With

U (�) = exp(i�a Qa),


where Qa are generators of G, we get

[Qa,�] = −T a� (1.178)

i.e. (1.23) with { , } replaced by −i[ , ].A group G of transformations in the space of internal quantum numbers is a

symmetry group provided the action is invariant U∫L(x)U−1 = ∫

L(x).

1.5 Discrete symmetries

Space reflection

Under the transformation of space reflection xµ → x ′µ with

t ′ = t, x′ = −x

the transformation laws for classical physical observables such as energy E , thethree-momentum p, total angular momentum J, helicity h = (J · p)/|p| etc. areobvious:

E ′ = E, p′ = −p, J′ = J, h′ = −h (1.179)

All internal charges are invariant with respect to space inversion.We may also address the question of the transformation law for classical fields.

For scalar fields, the most general form of transformation under space reflectionis:

�P(x′) =

{±�(x) for a real field

exp(iγ )�(x) for a complex field(1.180)

where the phase γ is arbitrary for a free field (a free scalar complex field has nowell-defined parity). Clearly, the Klein–Gordon equations of motion for scalarfields remain form-invariant under those transformations, i.e. they have the sameform in the original and in the primed reference frames.

The Weyl spinors transform under space inversion from the one in the ( 12 , 0)

representation of the Lorentz group to the one in the (0, 12) representation and vice

versa. Indeed, if two right-handed coordinate frames O1 and O2 are connectedby the Lorentz transformation with parameters ηi , ξi (see Appendix A for thedefinition of the parameters ηi and ξi ) then their images under space reflection,O′

1 and O′2 are connected by the Lorentz transformation with parameters ηi ,−ξi .

Correspondingly, a spinor λα transforming as λ′β = M(ηηη,ξξξ) αβ λα, seen from the

reflected frame, is a dotted spinor λαP transforming under Lorentz transformations

as:

λ′αP = [M†−1(ηηη,−ξξξ)]αβλβ

P , (1.181)

1.5 Discrete symmetries 49

since the matrices M†−1(ηηη,−ξξξ) and M(ηηη,ξξξ) are numerically equal. Thus:

λα(x) → λαP(x

′) ≡ exp(iγ )P αβλβ(x) (1.182)

where the only role of P ≡ σ 0 is to change the indices from undotted to dottedones. The transformation (1.182) satisfies the necessary relation:

[M†−1(ηηη,−ξξξ)]αβ

P βσ λσ (x) = P αβ[M(ηηη,ξξξ)] σβ λσ (x) (1.183)

because P is numerically equal to the unit matrix. Similarly, we get

χ α(x) → χPα(x′) ≡ exp(iγ )Pαβ χ

β (x) = exp(iγ )(σ 0)αβ χβ (x) (1.184)

and under Lorentz transformations χPα(x ′) transforms with M(ηηη,−ξξξ) =M†−1(ηηη,ξξξ).

The Weyl equation is not invariant under space reflection. The positive(negative) energy solution of the Weyl equation iσ µ∂µλ = 0 (see Appendix A):∼ exp(−ik · x)a(k) (∼ exp(ik · x)b(k)) upon substituting x = −x′ becomes thesolution with positive (negative) energy of the equation iσµ∂ ′µλ = 0. Thus, in thereflected coordinate frame it describes, in accord with (1.179), opposite helicitystate, h = + 1

2 (− 12 ), and transforms, therefore, according to the representation

(0, 12).

It is very important to notice that the parity transformation on the Weyl spinorstransforms fields ( 1

2 , 0) in representation R of a group G acting in the space ofinternal quantum numbers into (0, 1

2) again in representation R (and not R�).Therefore parity transformation of the Weyl spinors can be a symmetry of aphysical system only if it contains the corresponding pairs of fundamental fields.

For a four-component Dirac spinor in the chiral representation of the Dirac

matrices, � =(

λα

λcβ

), the transformation law follows directly from the trans-

formation of the Weyl spinors (1.182), (1.184).

�P(x′) =

(exp(iγ )(σ 0λc)α

exp(iγ )(σ 0λ)β

)(x) (1.185)

Such transformation preserves the form of the massive Dirac equation (i.e. �P(x ′)defined in (1.185) satisfies the Dirac’s equation in the reflected frame) provided wechoose γ = γ . In this case

�P(x′) = exp(iγ )γ0�(x) (1.186)

where the phase γ is still arbitrary. The space reflection is a symmetry of the freeDirac fermion. Moreover, bilinear forms like ��, �γ5�, �γµ�, �γµγ5� etc.have well-defined properties under the parity transformation.


The interaction fixes the relative parities of the different fields provided thatspace inversion is a good symmetry. If parity is not a symmetry of the lagrangian,there is no choice of the phases γ such that LP(�P) = L(�P). For instance,the pion–nucleon interaction can be described, assuming spin zero for the pion,by two Lorentz invariants �� and �γ5��. Thus, if parity is conserved instrong interactions, the pion field must be a scalar or a pseudoscalar (as seen instrong interactions). From more detailed study of strong interactions it has beendetermined experimentally that the pion field indeed has spin zero and negativeparity (and space inversion is a symmetry of strong interactions; see, for example,Werle (1966)).

In quantum theory, the Hilbert spaces H of the observer O and HP of theobserver OP (in the reflected coordinate frame) are isomorphic but can, in general,be different. For space reflection to be a symmetry of the physical system, it isnecessary that these two Hilbert spaces are identical and can be identified with eachother. This also means that the operators �P(x) acting in HP can be identified withsome operators acting in the original Hilbert space. We can then introduce a unitarylinear operator P which transforms the states from one reference frame to another,i.e. maps the Hilbert space onto itself, and study its action on the field operators.For complex scalar field theory, interpreting the transformation on the classicalfields as the transformation on the wave-functions, the analogue of (1.169) reads:

〈0|�P(x′)|vP〉 = exp(iγ )〈0|�(x)|v〉 (1.187)

(for a real scalar field exp(iγ ) = ±1) where |vP〉 = P|v〉.Identifying �P(x ′) ≡ �(x ′), this equation is satisfied with

P�(x)P−1 = exp(−iγ )�(x ′) (1.188)

(we assume the vacuum to be invariant under space reflection; if the invarianceunder space reflection is spontaneously broken we face the same situation as forthe spontaneously broken global symmetries) and for the creation and annihilationoperators we get

Pa(k)P−1 = exp(−iγ )a(−k)

Pb†(k)P−1 = exp(−iγ )b†(−k)

}(1.189)

Thus, for one-particle states we have

P|a, k〉 ≡ Pa†(k)|0〉 = exp(iγ )|a,−k〉P|b, k〉 ≡ Pb†(k)|0〉 = exp(−iγ )|b,−k〉

}(1.190)

The operator P can be expressed in terms of the creation and annihilation operators(see, for example, Bjorken & Drell (1965)).


For a chiral theory containing left-handed Weyl fields λiα transforming as a

complex representation R of the internal symmetry group the Hilbert spaces HP

and H mapped into each other by the parity transformation are, in general, different(the states in H and in HP have opposite helicities − 1

2 and + 12 , respectively).

Thus, space reflection cannot be a symmetry of such a theory. For this to bethe case, the system must also contain the right-handed field operators λci α(x),transforming as (0, 1

2) and R under the Lorentz and internal symmetry grouptransformations, respectively, with the expansions analogous to the ones given in(1.152) and (1.153):

λcα(x) =∫

d3k

(2π)32E

[bR(k)a′α(k) exp(−ik · x)+ d†

L(k)b′α(k) exp(ik · x)

]λc†α(x) =

∫d3k

(2π)32E

[b†

R(k)a′�α(k) exp(ik · x)+ dL(k)b′�α(k) exp(−ik · x)

]

(1.191)

where a′(k) and b′(k) are positive and negative energy chiral solutions to the freeWeyl equation iσµ∂µχ(x) = 0 and bR and d†

L are the annihilation and creationoperators. In other words, the Weyl fields must be in a real representation.For instance, R = R′ ⊕ R′� with, for example, λi

α transforming as R′ and λciα

transforming as R′�. The action of the parity operator P on the left-handed Weylfield λα(x) can now be defined. Identifying λα

P(x′) ≡ λcα(x ′) and following the

same arguments as for the scalar fields (see (1.187)), with (1.182) (or (1.184)) takenas the transformation law for the wave-functions, we get†

Pλα(x)P−1 = exp(−iγ )σ 0αβ

λcβ (x ′) (1.192)

Pλcα(x)P−1 = exp(−iγ )(σ 0)αβλβ(x′) (1.193)

For the creation and annihilation operators, using the properties (A.87) of theclassical solutions of the Weyl equations, we obtain

PbL(k)P−1 = bR(−k), Pd†R(k)P−1 = d†

L(−k) (1.194)

up to arbitrary and uncorrelated phases present in the classical solutions of theWeyl equations (A.61), (A.64).

For the four-component Dirac fields the parity operation can be realized as amapping of H onto itself. In this case we can identify the operator �P with �

(they have the same transformation properties under the internal symmetry group)

† If the representation R is irreducible and real (like, for example, adjoint representation of SU (N )) then onecan simply identify λαP (x ′) with SP λα (x ′), where SP is a constant matrix and parity can be a symmetry ofthe theory.


so that

P�(x)P−1 = exp(−iγ )γ 0�(x ′) (1.195)

and

Pb(p, s)P−1 = exp(−iγ )b(−p, s) (1.196)

Pd†(p, s)P−1 = − exp(−iγ )d†(−p, s) (1.197)

and similarly for b† and d but with the complex conjugate phase exp(iγ ). Thespinor properties γ 0u(−p, s) = u(p, s), γ 0v(−p, s) = −v(p, s) have beenused (see Appendix A). Analogous relations for helicity creation and annihilationoperators can be derived using (A.89) Although the phase factor exp(−iγ ) isarbitrary, the (massive) particle and its antiparticle always have negative relativeparity (assuming vacuum to be symmetric under the parity operation, P|0〉 =+|0〉):

P|bk1, s1, dk2, s2〉 ≡ Pb†(k1, s1)d†(k2, s2)|0〉

= −|b − k1, s1, d − k2, s2〉 (1.198)

Note that this conclusion does not hold for massless Weyl states because the phasesin (1.194) are arbitrary and uncorrelated (the states with opposite helicities are notrelated by Lorentz transformations).

For a massive Majorana field operator with the expansion given by (1.167) theapplication of the rule (1.195) gives two relations

Pb(k, s)P−1 = exp(−iγ )b(−k, s)

Pb†(k, s)P−1 = − exp(−iγ )b†(−k, s)

}(1.199)

which are compatible only if exp(−iγ ) = ±i. In this case the theory containingthe massive Majorana spinor may be parity-invariant. It is also worth noting thatfor such a particle

P|k, s〉 = ±i|−k, s〉i.e. a Majorana particle at rest (k = 0) is a parity eigenstate with purely imaginaryeigenvalue.

One can also check that the free Dirac lagrangian is invariant under spacereflection: PL(x)P−1 = L(x ′).

The parity of the electromagnetic field is determined by its coupling to theclassical current, jµ(x) = q�(x)γµ�(x). The requirement of the invarianceof the classical lagrangian Lint = −ejµ(x)Aµ(x) describing interaction of thecharged four-component Dirac field � with the electromagnetic field Aµ(x)


together with the property �P(x ′) = exp(iγ )γ0�(x) leads to the conclusion that�P(x ′)γµ�P(x ′)Aµ

P(x′) = �(x)γµ�(x)Aµ(x), provided

A0P(x

′) = A0(x), AP(x′) = −A(x) (1.200)

Thus, in quantum theory we identify

A0P(x

′) ≡ A0(x ′) = PA0(x)P−1

AP(x ′) ≡ A(x ′) = −PA(x)P−1

}(1.201)

Time reversal

The classical sense of time inversion is clear: the velocities of a system in somefinal state are reversed and it is moving backwards. If the dynamics is invariantunder such transformation, the system will eventually reach the original initialstate. Formally, time reversal can be described by the change of coordinates†xµ → x ′µ with

t ′ = −t, x′ = x (1.202)

Classical physical observables behave as follows (compare with (1.179)):

E ′ = E, p′ = −p, J′ = −J, h′ = h (1.203)

(all time derivatives change sign). All charges remain, of course, unchanged. Weconsider now a classical complex scalar field �(x). Its transformation undertime reversal can be inferred from the physical requirement that the charges(i.e. j a

0 (t, x)) do not change whereas for the space coordinates of the currentj ai (−t, x)T = − j a

i (t, x) (since the velocities are reversed). From (1.16) (or (1.57)for the abelian case) we get

�T (x′) = exp(iγ )��(x) (1.204)

where the phase factor can actually be only ±1 (since the real componentfields cannot change). Clearly, a positive energy plane wave with momentum ptransforms into a positive energy plane wave in the reflected frame with momentump′ = −p, in accord with (1.203). The Klein–Gordon equation is form-invariantunder time reversal, i.e. if (∂2 + m2)�(x) = 0 then also (∂ ′2 + m2)�T (x ′) = 0.If the fields carry internal quantum number index i and transform as �′(x) =exp(−i�aT a)�(x) then the fields �T (x ′) transform the same way in spite of thecomplex conjugation because the change t →−t assures invariance of the charges.The coupling with the electromagnetic field is invariant under the transformation

† Physically, it is however easier to think in terms of the active transformation.


(1.204) and the simultaneous transformation of the potential four-vector: A0(x ′) =A0(x), Ai (x ′) = −Ai (x).

Since helicity does not change under time reversal, the Lorentz transformationproperties for the Weyl fields should not change. Because positive energy solutionswith with momenta p must transform into positive energy solutions with −p,the transformation on the Weyl fields must involve their complex conjugation.Moreover, from the additional physical requirement that charges are invariant andthe space components of the current jµ(x) = λσ µλ change sign we conclude thatthe transformation λα → λTα must include the matrix σ 2. Taking proper accountof the Lorentz indices we finally conclude that

λTα(x′) = exp(iγ )(iσ 1σ 3) β

α (λ�(x))β (1.205)

where the phase factor exp(iγ ) is arbitrary. Furthermore, similarly to the spaceinversion, one can argue that a Lorentz transformation described by parametersηi , ξi , in the ‘time-reflected’ frame is described by the parameters ηi ,−ξi . Thus,the necessary relation

exp(iγ )(iσ 1σ 3) βα

(M(ηηη,ξξξ) κ

β λκ

)� = M(ηηη,−ξξξ) βα exp(iγ )(iσ 1σ 3) κ

β (λκ)�

(1.206)

is fulfilled because σ 1σ 3 M�(ηηη,ξξξ) = M(ηηη,−ξξξ)σ 1σ 3. It is obvious that λTα(x ′)satisfies the Weyl equation for the left-handed spinors in the primed variables: iσ ·∂ ′λT (x ′) = 0.

Similarly, the right-handed spinors transform into

χ α → χ αT (x

′) = exp(iγ )(iσ 1σ 3)αβ(χ β (x))� (1.207)

and χ αT (x

′) transform again as (0, 12) under the Lorentz transformations.

The transformation law for the four-component Dirac spinors follows directlyfrom the transformation properties of the left- and right-handed Weyl spinors. Inthe chiral representation of the Dirac matrices we have

�(x) → �T (x′) =

(exp(iγ )iσ 1σ 3λ�(x)

exp(iγ )iσ 1σ 3λc�(x)

)(1.208)

Such transformation preserves the form of the massive Dirac equation (i.e. �T (x ′)satisfies the Dirac equation in the primed variables) provided γ = γ . In this case(1.208) can be rewritten as

�T (x ′) = exp(iγ )iγ 1γ 3��(x)

�†T (x

′) = − exp(iγ )i�†�(x)γ 3γ 1

}(1.209)

with the phase γ still arbitrary.


In the quantum case we define an operator T such that |vT 〉 = T |v〉. As for thespace inversion, the time reflection can be a symmetry of the physical system onlyif the two Hilbert spaces can be identified with each other.

Interpreting the classical transformation (1.204) as the transformation on thewave-function 〈0|�(x)|v〉 for a state |v〉 we must have:

〈0T |�T (x′)|vT 〉 = exp(iγ )〈0|�(x)|v〉� (1.210)

It follows that the operator T must be antilinear (antiunitary): T †T = 1 andT (α1|v〉 + α2|w〉) = α�

1T |v〉 + α�2T |w〉. This can be seen by applying (1.210)

to a state vector |v〉 = α1|v1〉+α2|v2〉, with complex coefficients αi .† By applyingthe antiunitarity condition successively to |v〉, |v1〉 + |v2〉 and to |w〉 = |v1〉 + i|v2〉one can show that (Wigner 1931)

〈wT |vT 〉 = 〈v|w〉 = 〈w|v〉� (1.211)

The l.h.s. of (1.210) can now be written as

〈0T |�T (x′)|vT 〉 = 〈0T |T T −1�T (x

′)T |v〉 = 〈0|T −1�T (x′)T |v〉� (1.212)

where we have applied‡ the rule (1.211) to the state T (T −1�T (x ′)T |v〉). There-fore, identifying �T (x ′) with �(x ′) and comparing with the r.h.s. of (1.210), weconclude that

T �(x)T −1 = exp(−iγ )�(x ′) (1.213)

(for a real scalar field we must have exp(−iγ ) = ±1). The transformation on thequantum fields does not involve any complex conjugation and the consistency withthe classical transformation is ensured due to the antilinearity of T . From (1.213)for the creation and annihilation operators we get:

T a(k)T −1 = exp(−iγ )a(−k), T b†(k)T −1 = exp(−iγ )b†(−k) (1.214)

and similarly for their hermitean conjugates.For the Weyl fields we can identify HT with H and the operators λTα with λα.

The equation for the matrix elements similar to (1.212) leads to

T λα(x)T −1 = exp(−iγ )(iσ 1σ 3) βα λβ(x ′)

T λα(x)T −1 = exp(−iγ )(iσ 1σ 3)αβλβ (x ′)

(1.215)

† Antiunitarity of the T operator can be justified also by applying it to the time component of (1.174):

T [H,�(x)]T −1 = T(−i

∂�(x)

∂t

)T −1

The physically motivated requirement that T H(t)T −1 = H(t ′) again leads to the conclusion that if �(x ′) isto satisfy (1.174) with primed variables then T must be antiunitary.

‡ Notice that since T is antiunitary 〈0T | �= 〈0|T −1.


Using the properties (A.91) of the solutions to the Weyl equations we get for thecreation and annihilation operators

T bL(k)T −1 = exp(−iγ )bL(−k), T d†R(k)T −1 = exp(−iγ )d†

R(−k) (1.216)

(up to some arbitrary phase factors associated with the classical solutions for theWeyl spinors) and similar relations for the hermitean conjugate operators. It is clearthat the operator T changes the direction of the momentum of the state leaving itshelicity unchanged.

For the four-component Dirac fields one gets

T �(x)T −1 = exp(−iγ )(iγ 1γ 3)�(x ′) (1.217)

leading, upon using the properties of the classical solutions to the Dirac equation(summarized in (A.93)), to the relations

T b(k, s)T −1 = −i(−1)s exp(−iγ )b(−k, s ′)

T d†(k, s)T −1 = i(−1)s exp(−iγ )d†(−k, s ′)

}(1.218)

(s ′ = 2(1) for s = 1(2); s = 1(2) denotes the spin projection equal + 12(− 1

2) withrespect to the chosen quantization axis). For the massive Majorana field (1.167)the application of rule (1.217) leads to the first relation (1.218) and the consistencycondition exp(−2iγ ) = exp(2iρ).

The transformation properties of the electromagnetic field under the operation oftime reversal (already discussed) can be determined by its coupling to the classicalcurrent, jµ(x) = q�(x)γµ�(x). As in the case of the parity transformation, therequirement of the invariance of the lagrangian Lint = −ejµ(x)Aµ(x) togetherwith the properties (1.209) leads to the conclusion† that �T (x ′)γµ�T (x ′)Aµ

T (x′)

= �(x)γµ�(x)Aµ(x) provided

A0T (x

′) = A0(x), AT (x′) = −A(x) (1.219)

Thus, in quantum theory

A0T (x

′) ≡ A0(x ′) = T A0(x)T −1

AT (x ′) ≡ A(x ′) = −T A(x)T −1

}(1.220)

Charge conjugation

The transformation of charge conjugation is genuinely connected to quantumphysics. It can be defined as a transformation on wave-functions (positive energysolutions to the classical wave equations) of quantum systems. We consider a

† In verifying the invariance of the quantum lagrangian by using (1.218) one should also remember thatT γµT −1 = γµ�.


physical system which carries some internal quantum numbers (charges) and isdescribed by the wave-function �i (x) transforming as a representation R of agroup of transformations G:

abelian �′pos(x) = exp(−iq�)�pos(x) (1.221)

or

non-abelian �i ′pos(x) =

(exp(−i�aT a)

)i

j� j

pos(x) (1.222)

We assume that the group G is a global symmetry group of this system. Onecan introduce a ‘charge conjugate’ reference frame (a ‘reflection’ in the space ofinternal charges). For an observer using this reference frame the same physicalsystem is described by another wave-function �c

i (x) which transforms as repre-sentation R� of G, complex conjugate to R but has the same space-time Lorentztransformation properties and the same dependence on the space-time variables asthe wave-function �i (x)

�c′pos(x) = exp[−i(−q)�]�c

pos(x) (1.223)

or

�ci ′pos(x) =

(exp[−i�a(−T a�)]

)i

j�cj

pos(x) (1.224)

(see (E.10) for the definition of complex and real representations). Thus, ‘opposite’internal charges are assigned to the same system. This change of the ‘referenceframe’ is called the transformation of charge conjugation (we could as well take theactive view and replace the original system by a new one with ‘reversed’ charges).This transformation may or may not be a symmetry of a given system.

Charge conjugation is a symmetry of the system if the charge conjugate wavefunction �c

i (x) can be identified with a solution of the relativistic equationof motion of the system, obtained by a transformation on its negative energysolutions and usually also called charge conjugation. One should stress thatfrom the fundamental fact of the existence of the negative energy solutions tothe relativistic wave equations follows the existence of particle and antiparticlepairs (i.e. the states whose wave-functions transform as R and R�, respectively):the negative energy solutions can always be reinterpreted as antiparticle (positiveenergy) wave-functions. However, after such reinterpretation (often and somewhatmisleadingly also called charge conjugation) the antiparticle wave-functions maysatisfy different equations of motion than the negative energy solutions did andcannot be identified with the ‘charge conjugate’ wave-function of the system.This means that the system seen by the observer in the new reference framesatisfies other equations of motion than in the original reference frame and thetransformation of charge conjugation is not a symmetry of this particular system.


Again, this does not change the fact that particles and antiparticles do exist in pairs,as predicted by the presence of the negative energy solutions to the equations ofmotion.

We now consider several examples. We begin with a set of free complex scalarfields �i (x) which satisfy the Klein–Gordon equation. This equation is invariantunder the operation

�i (x) → �ci (x) ≡ exp(iγ )��

i (x) (1.225)

Thus, if �i (x) are solutions of the Klein–Gordon equation the same is true for thefunctions �c

i (x). It is trivial but important to notice that the operation (1.225)transforms negative energy solutions into positive energy ones and vice versa,i.e. transforms solutions �

negi (x) which have no wave-function interpretation into

�c posi (x) which can be interpreted as wave-functions of antiparticles (with reversed

global quantum numbers). Moreover, the space-time dependences of the functions�

posi (x) and �

c posi (x) are the same. Thus, the existence of the symmetry operation

(1.225) for the relativistic scalar wave equation provides the interpretation of itsnegative energy solutions and one predicts the existence of particle–antiparticlepairs described by wave-functions with the same space-time form but oppositeinternal charges.

We may extend our considerations to scalar fields with gauge symmetries, i.e.coupled in a gauge-invariant way to (abelian or non-abelian) gauge vector fields.The equations of motion follow then from the lagrangian

L = ∣∣(∂µ + ig AaµT a)�∣∣2

= �∗(←∂µ − ig AaµT a

)(→∂µ + ig AaµT a

)�

(in going from the first to the second line the hermiticity of the generators T a hasbeen used). It is easy to check that they are invariant under the operation

�i (x) → �ci (x) = exp(iγ )(�i (x))�

Aaµ(x) → Aca

µ (x) : Acaµ (x)(T a)∗ = −Aa

µ(x)Ta

}(1.226)

i.e. the wave-function of the antiparticle in the charge conjugate vector potentialAca

µ (x) has the same form as the particle wave-function in the potential Aaµ(x).

For the Weyl equation there is no symmetry operation which transforms itsnegative energy solutions λ

negiα (x) (λα neg

i (x)) into solutions λc posiα (x) (λcα pos

i (x)) ofthe same equation with the same Lorentz transformation properties and the samespace-time dependence as λpos

iα (x) (λα posi (x)) and with the internal charges reversed

compared to the charges of λposiα (x). Instead, the operation λiα(x) → λα

i (x)transforms the negative energy solution λ

negiα (x) (transforming as R and ( 1

2 , 0) under


internal and Lorentz groups, respectively) into λα posi (x) which satisfies the Weyl

equation for right-handed spinors (i.e. transforming as R� and (0, 12)).

The same conclusion remains valid for the Weyl fields coupled to the abelian ornon-abelian gauge fields. Therefore the positive energy solution to a Weyl equationis a wave-function of a particle with h = − 1

2(+ 12) and the ‘charge conjugate’

form of the negative energy solution is the wave-function of its antiparticle withh = + 1

2(− 12).† However, for a quantum system described by a set of left- and

right-handed Weyl equations with solutions λposiα (x), λcα pos

i (x) transforming as R,( 1

2 , 0) and R�, (0, 12) there exists the symmetry operation λiα → λα

i , λcαi → λc

iα

which transforms the negative energy solutions of each of the equations into thepositive energy solutions of the other equations, with the same Lorentz and space-time dependence as λ

α posi (x) and λ

posiα (x), respectively. Such a system is equivalent

to a Dirac field

�(x) =(

λiα(x)

λcβi (x)

)(1.227)

which satisfies the Dirac equation. The operation

�(x) → �c(x) = exp(iγ )

(λc

iα(x)

λβ

i (x)

)(1.228)

transforms the negative energy solutions �neg(x) into positive energy �c pos(x)which transform as R� and have the same space-time dependence as �pos(x). Fora general Dirac field (a solution to the massive Dirac equation) there exists thesymmetry operation

�c = exp(iγ )Cγ 0�� = exp(iγ )C�T ,

�c = − exp(−iγ )�T C−1

}(1.229)

where the 4×4 matrix C (its explicit form and properties are given in Appendix A)satisfies (Cγ0)γ

µ�(Cγ0)−1 = −γ µ, so that �c indeed satisfies the Dirac equation.

For the four-component spinors we have CuT (p, s) = v(p, s) and C vT (p, s) =u(p, s).

For a Dirac field coupled to (abelian or non-abelian) gauge fields the lagrangian

L = �(i/∂ − g A/aT a − m

)�

is invariant under simultaneous transformation

� → �c, Aaµ → Aca

µ (x) (1.230)

with Acaµ (x) defined in (1.226) (remember that �s are anticommuting c-numbers;

† Note that the helicity of the solution λα posi (x) is the same as the helicity of λneg

iα (x).


the Dirac lagrangian is invariant up to a total derivative). After such transforma-tions, the space-time dependence of the solutions �(x) and �c(x) remains thesame.

With few exceptions (see (1.243)), the arbitrary phase γ in (1.225) and (1.229)is unphysical and can be taken as zero. The transformation properties of the spinorbilinears follow from (1.229): �c

1�c2 = �2�1, �c

1γµ�c2 = −�2γµ�1 etc.

Moving now to the description of the physical system in terms of the statevectors (rather than their representation in the configuration space) the operationof charge conjugation maps the original Hilbert space H onto the Hilbert space Hc

which describes the physical system in the ‘charge conjugate’ frame. The statesin Hc have the same space-time description but opposite internal charges. Thisoperation can be a symmetry only if H and Hc are identical and can be identifiedwith each other (in consequence, the operators acting in H can be identified withsome operators acting in Hc). We can then define a linear (unitary) operator C,|vc〉 = C|v〉. Next, we would like to find the action of C on the field operators. Weuse the fact that for the wave-functions 〈0|�c(x)|vc〉 = exp(iγ )〈0|�(x)|v〉, i.e.they have the same space-time dependence. For scalar field theory, we see that, inagreement with the transformation properties under internal symmetry group, wecan identify the operators

�c(x) ≡ �†(x) (1.231)

getting†

C�i (x)C−1 = exp(−iγ )�†i (x) (1.232)

The presence in (1.232) of an arbitrary phase reflects the freedom to redefine anyquantum state by multiplying it by a phase factor. In the presence of interactionsthis freedom can be used to define the maximal discrete symmetry of the system.The creation and annihilation operators defined in (1.126) and (1.143) satisfy thefollowing relations

Ca(k)C−1 = ±a(k) (1.233)

for real scalar fields (1.126) and

Ca(k)C−1 = exp(−iγ )b(k), Cb†(k)C−1 = exp(−iγ )a†(k) (1.234)

for complex scalar fields (1.143). Remembering our discussion in Section 1.4 wesee that, consistently with its definition, charge conjugation transforms a particlestate with four-momentum (E, k) into the antiparticle state with the same four-momentum.

† It is worth stressing the difference between charge conjugation and time reversal, compare (1.225) with (1.204)and (1.213) with (1.232). This difference follows from the antilinearity of T .


It is sometimes very useful to consider the eigenstates of the charge conjugationoperator C (similar comments apply to the combined transformation C P). Forinstance, when C (and/or C P) is a good symmetry and the system is neutral in allconserved charges (the latter is necessary since, for example, for abelian chargesCQ = −QC, so no common eigenstates exist) then the description in terms of the C(or CP) eigenstates is convenient. Important physical examples are the positroniumstate (electromagnetic interactions conserve both C and Q but positronium iselectrically neutral) and the K0–K0 (or B0–B0) mixing by electroweak interactions(CP is almost conserved but the flavour charges like strangeness or beauty are not).The K0–K0 mixing will be discussed in more detail at the end of this section andin Section 12.7. Assuming the vacuum to be invariant under C , it then follows thatthe states given by the linear superposition of the particle and antiparticle states

|kC〉 = exp(iφ±)√2

(a†(k)|0〉 ± exp(iγ )b†(k)|0〉) (1.235)

are eigenvectors of the charge conjugation operator, with eigenvalues ±1, respec-tively. The presence in this equation of the phase factor exp(iγ ) is crucial for aphase-convention-free conclusion. Note also that the state |kC〉 can be written as

|kC〉 = exp(iφ±)√2

(a†(k)|0〉 ± Ca†(k)|0〉) (1.236)

In a very similar manner one can construct the states at rest invariant under the C Ptransformation even when C and P are not conserved separately. For the stateswith momentum k the C P acts as follows: CP|a, k〉 = exp(iγ )|b,−k〉.

For the Weyl fields, the Hilbert space HC can be identified with H (and Ccan be a symmetry operations) only if the set of those fields transforms as areal representation of the group G. With λα and λc

α transforming as R and R�,respectively, and following the same arguments as for the (1.232) we get

CλαC−1 = exp(−iγ )λcα (1.237)

and a similar equation for λ†α with the opposite phase. Using the expansions of the

fields λα and λcα given in (1.152) and (1.153) we arrive at:

CbL(k)C−1 = exp(−iγ )dL(k), (1.238)

Cd†R(k)C−1 = exp(−iγ )b†

R(k), (1.239)

up to arbitrary phases present in the classical solutions for the Weyl equations.It is clear that a particle state with momentum k and helicity h is transformedunder charge conjugation into an antiparticle state with the same momentum andhelicity.


Following the same arguments, the action of C on the four-component Dirac fieldoperators is given by

�c(x) ≡ C�† = exp(iγ )C�(x)C−1 (1.240)

which leads to

Cb(k, s)C−1 = exp(−iγ )d(k, s) (1.241)

Cd†(k, s)C−1 = exp(−iγ )b†(k, s) (1.242)

for the creation and annihilation operators. We have used the transformationproperties (A.46) for spinors u(k, s) and v(k, s).

For a Majorana field with expansion (1.167), using the definition (1.229) weobtain

Cb†(k, s)C−1 = exp(−iρ) exp(−iγ )b†(k, s)

Cb(k, s)C−1 = exp(−iρ) exp(−iγ )b(k, s)

}(1.243)

Consistency of these two equations requires

exp(−iρ) exp(−iγ ) = ±1 (1.244)

Thus, the Majorana field satisfies

C�M(x)C−1 = ±�M(x) (1.245)

It is its own charge conjugate up to a sign. Also, it follows that a Majorana particleis an eigenstate of the charge conjugation operator with eigenvalue ±1. Thus, astate of such a particle at rest may be a CP eigenstate with eigenvalues ±i (see(1.199)).

Summary and the C PT transformation

We have discussed the action of the discrete transformations (space, time andcharge inversions) on classical fields. We have identified the requirements whichare necessary for these transformations to be symmetries of free (or interacting withthe gauge fields) quantum systems of scalar, Weyl and Dirac particles describedby state vectors in some Hilbert space. For those transformations which aresymmetries of the system one can introduce unitary operators acting on quantumfields.

It is interesting that even free particle systems are not all invariant under thethree discrete transformations. We have learned that a system of free Weylparticles which transforms as a complex representation R of some group ofinternal symmetries is invariant under the time reflection but not under separate


transformations of space reflection and charge conjugation. It is natural to ask whathappens to such a system under simultaneous C and P transformations. It turnsout that the Weyl equation is invariant under the replacement λα(x) → λC P(x) =exp(−iγ )λα(t,−x), i.e. the field λC P(x) satisfies the same equation as the fieldλα(x). For quantized fields CPλα(x)(CP)−1 = exp(−γ )λα(t,−x) and then using(1.152) and (1.153) we can infer the transformation properties of the annihilationand creation operators:

CPbL(k)(CP)−1 = exp(iγ )dR(−k)

C Pd†R(k)C P−1 = exp(iγ )b†

L(−k)

}(1.246)

A particle state with momentum k and helicity h is transformed under C Ptransformation into an antiparticle state with opposite momentum and helicity (asexpected from the Dirac sea interpretation of the negative energy solutions to theWeyl equation). We conclude that all the considered systems are invariant underthe C P and C PT transformations.

The discrete symmetries can easily be broken by interactions or even mass terms.Consider the theory of Weyl fermions in which each set of λs transforming as Rhas its counterpart λc transforming as R� of the internal symmetry group. The mostgeneral hermitean mass term of the lagrangian is

Lmass = −m ABλAλcB − m�

AB λAλcB (1.247)

It is straightforward to check that if Lmass is to conserve P we must have m AB =m�

B A, i.e. the mass matrix m AB must be hermitean. Similarly, for C to be conservedthe mass matrix must be symmetric. For C P to be conserved m AB must bereal. Finally, it can be checked that Lmass remains invariant under the C PTtransformation

CPT Lmass(CPT )−1 = Lmass (1.248)

independently of the form of the matrix m AB . Similar considerations apply tocomplex Yukawa couplings. One should also stress that often there exists somearbitrariness in the field phase conventions, which can be used to make the maximalsymmetry of the lagrangian explicit.

The importance of the C PT transformation follows from a fundamental theoremof quantum field theory which says that C PT invariance follows from the basic re-quirements of locality, Lorentz invariance and unitarity. Among the consequencesof the C PT theorem are equal masses for particles and their antiparticles, equallifetimes, equal and opposite electric charges, and equal magnetic moments.

A number of experimental upper limits on violation of the C PT theorem canbe found in the Particle Data Book (1996). They include several measurements


on electrons and positrons, muons and protons. The most stringent bound existsfor the mass difference of the K0 and K0: mK/mK < 9 × 10−19. In spite of thetremendous accuracy with which the C PT theorem has been verified, it is of greatinterest if the theorem remains valid at even smaller scales of length, down to thePlanck scale. (For a review of speculations on the possibility of violation of theC PT theorem, see Ellis, Mavromatos & Nanopoulos (1996).)

C P violation in the neutral K0–K0-system

In this subsection, we shall discuss the effects of the C P violation in the K0–K0 mixing. The physical processes under consideration here will be neutral kaondecays to two pions. They were the first processes in which violation of the C Psymmetry was observed (Christenson et al. 1964). More details about C P violationin the neutral kaon system can be found in Jarlskog (1989) and Bernabeu (1997).

We recall that both pions and neutral kaons belong to the lightest quark–antiquark bound states (mesons) which are stable with respect to strong interac-tions. They decay only due to electroweak interactions. Their flavour structure isthe following:

|π+〉 = |ud〉 (1.249)

|π−〉 = |du〉 (1.250)

|π0〉 = 1√2

[|uu〉 − |dd〉] (1.251)

|K0〉 = |ds〉 (1.252)

|K0〉 = |sd〉 (1.253)

While the pion states π+, π− and π0 are physically observable particles, theneutral kaon states K0 and K0 combine to the following physical ones:

|KL〉 = 1[2(1 + |ε|2)]1/2

[(1 + ε)|K0〉 − (1 − ε)|K0〉] (1.254)

|KS〉 = 1[2(1 + |ε|2)]1/2

[(1 + ε)|K0〉 + (1 − ε)|K0〉] (1.255)

We choose our phase convention for normalization of the neutral kaon states sothat CP|K0〉 = |K0〉 and CPT |K0〉 = |K0〉.

The names of KL(ong) and KS(hort) originate from their drastically differentlifetimes τL � 5 × 10−8 s and τS � 9 × 10−11 s, respectively. On the otherhand, their masses mL � mS � 498 MeV are almost equal: m ≡ mL − mS �3.5 × 10−12 MeV.


If C P was conserved in Nature, the parameter ε would vanish. Then, KL andKS would be C P-odd and C P-even, respectively. In such a case, KL could notdecay to two pions which are C P-even in the S-wave. In reality, KL indeed decaysmuch more often to three than to two pions. This explains its long lifetime, becausedecays to three pions are strongly phase-space suppressed.

Let us consider a state which at the initial time t0 is a neutral kaon at rest. Fort > t0, we write it as a linear combination of eigenstates of the strong interactionhamiltonian Hs

|�(t)〉 = a(t)|K0〉 + a(t)|K0〉 +∑

n

bn(t)|n〉 (1.256)

where

Hs|K0〉 = mK|K0〉, Hs|K0〉 = mK|K0〉, Hs|n〉 = En|n〉 (1.257)

and bn(t = t0) = 0. The states |n〉 stand for all the eigenstates of Hs which aredifferent from |K0〉 and |K0〉.

The state |�(t)〉 satisfies the Schrodinger equation

id

dt|�(t)〉 = (Hs + Hw)|�(t)〉 (1.258)

where Hw is the electroweak interaction hamiltonian. Solving this equationperturbatively in Hw, we find the time evolution equation for a(t) and a(t)

id

dt

[a(t)a(t)

]=[

H11 H12

H21 H22

] [a(t)a(t)

](1.259)

expressed in terms of matrix elements of Hw between strong interaction eigen-states:

Hi j = Mi j − i

2�i j (M = M†, � = �†) (1.260)

Mi j = mKδi j + 〈i |Hw| j〉 + P∑

n

〈i |Hw|n〉〈n|Hw| j〉mK − En

+O(H 3w) (1.261)

�i j = 2π∑

n

〈i |Hw|n〉〈n|Hw| j〉δ(mK − En)+O(H 3w) (1.262)

Here, the sum over n denotes the sum-integral over all states different from |K0〉 ≡|1〉 and |K0〉 ≡ |2〉, while P stands for the principal value.†

The matrix Hi j in (1.259) is not hermitean, because the coefficients bn(t) havebeen removed from the evolution equation. This is the reason why the absorptivepart �i j arises in Hi j . It parametrizes the decay rates of the kaons to other stronginteraction eigenstates.

† Eqs. (1.261) and (1.262) are obtained by sending t0 to (−∞) and turning Hw adiabatically off at t → ±∞(see, for example, Landau & Lifshitz, (1959) §43).


It is important to note that the O(Hw) and O(H 2w) parts of Hi j are actually of

the same order in the electroweak coupling constants. This is because the leadingcontribution to 〈i |Hw| j〉 originates from double W± boson exchange (Fig. 12.17),while kaon decays can proceed via only single W± boson exchange.

The C PT symmetry of electroweak interactions implies that the diagonalelements of the matrix Hi j are equal, i.e.

H11 = H22 ≡ H (1.263)

If electroweak interactions were symmetric under C P , then the off-diagonalelements H12 and H21 would be equal, too. Although the latter equality is violatedin Nature, it is usually convenient to verify what happens in this limit with therelations we shall discuss below.

The eigenvalues of the matrix Hi j are

λL,S = mL,S − i

2�L,S = H ± H 1/2

12 H 1/221 (1.264)

while the corresponding eigenvectors |KL〉 and |KS〉 are given in (1.254) and(1.255), respectively. The parameter ε which occurs in these equations is related tothe matrix Hi j as follows:

ε = H 1/212 − H 1/2

21

H 1/212 + H 1/2

21

. (1.265)

Here, we have assumed that H 1/221 tends continuously to H 1/2

12 when H21 tends toH12 in the limit of no C P violation.†

Once we know the eigenbasis of Hi j , solving (1.259) is trivial. The consideredstate |�(t)〉 can now be written as follows:

|�(t)〉 = cL exp(−imLt − 1

2�Lt) |KL〉

+ cS exp(−imSt − 1

2�St) |KS〉 +

∑n

bn(t)|n〉 (1.266)

with some time-independent coefficients cL and cS. Thus, mL.S and �L,S are,respectively, masses and decay widths of the physical neutral kaons KL and KS.

The observable we would like to consider is the ratio of decay amplitudes of KL

and KS into two-pion states with vanishing total isospin

εK = A[KL → (ππ)I=0]

A[KS → (ππ)I=0](1.267)

† This may not hold only in some really peculiar phase conventions for the parametrization of the Cabibbo–Kobayashi–Maskawa matrix and/or for the complex square root.


where

|(ππ)I=0〉 = 1√3

(|π+π−〉 + |π−π+〉 − |π0π0〉) (1.268)

It is convenient to express εK in terms of the decay amplitudes

A[K0 → (ππ)I=0] ≡ A0 exp(iδ0)

A[K0 → (ππ)I=0] ≡ A0 exp(iδ0)

}(1.269)

where δ0 is the strong phase shift in elastic scattering of two pions in the isospinzero state. The C PT symmetry implies that A0 = A∗

0. Using this fact together with(1.254), (1.255), (1.265), (1.267) and (1.269), one finds after a short calculationthat

εK = H 1/212 A0 − H 1/2

21 A∗0

H 1/212 A0 + H 1/2

21 A∗0

(1.270)

which can be equivalently rewritten as

εK = H12 A20 − H21 A∗2

0

4H 1/212 H 1/2

21 |A0|2(1 − ε2

K) =H12 A2

0 − H21 A∗20

[2(mL − mS)− i(�L − �S)]|A0|2 (1 − ε2K)

(1.271)The advantage of the latter formula is that it is manifestly independent of signconventions used for complex square roots and of phase conventions in theparametrization of the Cabibbo–Kobayashi–Maskawa (CKM) matrix.

Now, it is time to make some approximations. We know from experiment thatεK � 1 and �L � �S � 2(mL − mS) ≡ 2 m. Consequently,

εK � 2i Im(M12 A20)+ Im(�12 A2

0)

2(1 + i) m|A0|2 , (1.272)

where we have used the decomposition (1.260) of Hi j into hermitean and antiher-mitean parts Mi j and �i j , respectively.

Next, we take into account the experimental fact that KS decays dominantly toisospin-zero two-pion states. This means that the sum representing �i j in (1.262)is dominated by (ππ)I=0 intermediate states. Consequently, �12 is approximatelyproportional to A∗

0 A0 = A∗20 , and Im(�12 A2

0) can be neglected. Therefore,

εK � Im(M12 A20)√

2 m|A0|2 exp(iπ/4). (1.273)

In general, the phase of the amplitude A0 can take any value between 0 and 2π ,depending on phase conventions in the parametrization of the CKM matrix. In the


standard phase convention (see Chapter 12), A0 is approximately real, and we canwrite

ε � εK � Im M12√2 m

exp(iπ/4). (1.274)

The equality of ε and εK for real A0 follows directly from comparing (1.270) and(1.265). However, the parameter ε is, in general, dependent on phase conventions.Therefore ε is not an observable, unlike εK.

The value of εK is measured quite precisely nowadays, and it is of order 10−3 (forthe description of some of the methods of measuring εK see Marshak, Riazuddin& Ryan (1969). On the other hand, ε does not even need to be small. When it issmall, we can write in analogy to (1.272)

εK � 2i Im M12 + Im�12

2(1 + i) m(1.275)

However, we cannot neglect Im�12 before specifying the phase convention. Aswe have observed before, the phase of �12 is approximately opposite to the phaseof A2

0. On the other hand, as follows from (1.273), the quantity Im(M12 A20) is a

measure of C P violation in the neutral kaon mixing, Thus, if εK differs from zero,then either Im M12 or Im�12 must be non-vanishing. Consequently, the value of εis bounded from below when C P is violated. In other words, when C P is violated,then the mass eigenstates KL and KS in (1.254) and (1.255) are not C P eigenstates.

Problems

1.1 Consider the action for a motion of a particle in a potential V (x):

S([x], t1, t2) =∫ t2

t1dt

[1

2m

(dxdt

)− V (x)

]where [x] denotes a functional of the path x(t).

Calculate the variation S under the transformation

x(t) → x(t)+ δx(t)

Derive the Newton equation m d2x/dt2 = −∇∇∇V from the principle of the minimalaction (subject to the constraint δx(t1) = δx(t2) = 0). Prove that energy E ,momentum p and orbital momentum L = r×p are constants of motion if the potentialis symmetric under translations and rotations.

1.2 Write down the Newton equation d2x/dt2 + ∂V/∂x = 0 in the new reference frame,corresponding to the transformation t ′ = t + εt2. Construct the lagrangians whichgive both equations and compare with (1.27).

Problems 69

1.3 Use (1.35) to study conservation laws in classical electrodynamics (in the absence ofsources) with L = − 1

4 Fµν Fµν , where Fµν = ∂µ Aν − ∂ν Aµ. Derive the canonicalenergy–momentum tensor and bring into the symmetric and gauge-invariant form

�µν = (Fµα Fν

α + 14 gµν Fαβ Fαβ

)Check that it is traceless. Express �µν in terms of the electric E and magnetic B fields.In particular show that the energy density is �00 = 1

2 (E2 + B2) and the momentum

density �0i = (E×B)i is the Poynting vector. The conservation law ∂0�00+∂i�

0i =0 connects the change in the energy density to the Poynting vector. Finally, derive theangular momentum vector for the electromagnetic field and split it into orbital andintrinsic orbital momentum parts.

1.4 Consider a closed system of a charged classical particle interacting with an electro-magnetic field. Write down the action in terms of the electromagnetic potential Aµ

and derive the equations of motion and the energy–momentum conservation laws.

1.5 Show that for an abelian gauge theory the Noether current corresponding to the gaugesymmetry transformation (1.91) and (1.94) is

Sµ = ∂ρ(Fρµ�(x))

and that the associated conserved charge

Qs =∫

d3x S0(x, t)

either vanishes or reduces to the electric charge associated with global U (1) invarianceprovided � approaches an angle-independent limit at spatial infinity. Thus gaugeinvariance leads to no new conservation laws as compared to global invariance.

1.6 In abelian gauge theory the dual field strength tensor Fµν is defined as

Fµν = 12ε

µνρσ Fρσ , ε0123 = −1

Show that Fµν Fµν is a total divergence

Fµν Fµν = ∂µKµ

where

Kµ = εµνρσ Aν Fρσ

In a non-abelian case a matrix-valued dual field strength is

Gµν = 12ε

µνρσ Gρσ

and

Tr[Gµν Gµν] = ∂µKµ

where

K ρ = ερσµν Tr[Gσµ Aν + 23 Aσ Aµ Aν]


1.7 Check the Jacobi identity for the covariant derivatives

[Dµ, [Dν, Dρ]] + [Dν, [Dρ, Dµ]] + [Dρ, [Dµ, Dν]] = 0

and the Bianchi identity

DµGνσ + DνGσµ + Dσ Gµν = 0

or

DµGµν = 0

1.8 In pure Yang–Mills theory the Noether current for global symmetry transformations is

Y jaµ = cabc Ab

νGcνµ

and for gauge transformations

Y jµ� = 2

g2Tr[GνµDν�]

Using field equations DµGµν = 0 show that both currents are conserved. Show thatthe charge

Qa =∫

d3x Y ja0 (x, t)

is time-independent, provided Yja falls off sufficiently rapidly at large |x|. Since Yja isnot gauge-invariant, the fall-off requirement restricts the large |x| behaviour of gaugetransformation. Show that Qa is gauge-covariant against gauge transformations whichapproach a definite angle-independent limit as |x| → ∞. Show that the current Y jµ�generates no new charges provided � approaches an angle-independent limit at spatialinfinity.

Include fermion fields. Using Noether’s theorem derive the conserved symmetrycurrent:

J aµ = ja

µ + cabc AbνGcν

µ

where jaµ is given by (1.123). Show that the ordinary conservation of J a

µ is equivalentto the covariant conservation of ja

µ and the gauge field equation (1.124). Show that

∂µGµνa = g J ν

a

2

Path integral formulation of quantum field theory

2.1 Path integrals in quantum mechanics

Our first aim is to show that the matrix elements of quantum mechanical operatorscan be written as functional (path) integrals over all trajectories, with the integranddependent on the action integral. Intuitively, the need for integration is obvious:the position operator does not commute with the momentum or the hamiltonian.Consequently, time evolution changes the position eigenstate into one in whichposition is not determined. The quantum system has no definite trajectory and itis necessary to take a sum over all possible ones, according to the superpositionprinciple.

Transition matrix elements as path integrals

Consider a quantum mechanical system with one degree of freedom. The eigen-states of the position operator are introduced as follows

XH(t)|x, t〉 = x |x, t〉 Heisenberg picture

XS|x〉 = x |x〉 Schrodinger picture

with the relation

|x〉 = exp[−(i/h)Ht]|x, t〉where H denotes the hamiltonian of the system. The matrix element

〈x ′, t ′|x, t〉 = 〈x ′| exp[−(i/h)H(t ′ − t)]|x〉 (2.1)

corresponds to the transition from the eigenstate |x〉 at the moment of time t to thestate |x ′〉 at the time t ′, and is a Green’s function: define |t〉 by H |t〉 = ih(∂/∂t)|t〉,then

〈x |t ′〉 =∫

dx ′ 〈x | exp[−(i/h)H(t ′ − t)]|x ′〉〈x ′|t〉

71

72 2 Path integrals

The matrix element (2.1) we shall first represent as a multiple integral which willthen be used to define the functional integral by a limiting procedure. First wedivide the time interval (t ′ − t) into (n + 1) equal parts of length ε

t ′ = (n + 1)ε + t

t j = jε + t ( j = 1, . . . , n)

Next, we use the completeness relation at each of the times t j :∫dx j |x j , t j 〉〈x j , t j | = 1 (2.2)

together with

〈x j , t j |x j−1, t j−1〉 =⟨x j

∣∣∣∣ exp

(− i

hεH

)∣∣∣∣x j−1

⟩= 〈x j |x j−1〉 − iε

h〈x j |H |x j−1〉 + O

(ε2)

(2.3)

where x0, xn+1, t0, tn+1 are to be understood as x, x ′, t, t ′, respectively. Choosingthe hamiltonian H = H(P, X) to be of the form H = f (P)+ g(X) we can write

〈x j |H |x j−1〉 =∫

dp j 〈x j |p j 〉〈p j |H |x j−1〉

=∫

dp j

2π hexp

[i

hp j (x j − x j−1)

]H(p j , x j−1) (2.4)

where H(p, x) is now the classical c-number hamiltonian. Using (2.4), (2.3)becomes

〈x j , t j |x j−1, t j−1〉=∫

dp j

2π hexp

[i

hp j (x j − x j−1)

][1 − i

hεH(p j , x j−1)

]+ O

(ε2)

=∫

dp j

2π hexp

[i

hp j (x j − x j−1)− i

hεH(p j , x j−1)

]+ O

(ε2)(2.5)

and we obtain the following expression for the matrix element (2.1):

〈x ′, t ′|x, t〉 = limn→∞

∫ n∏j=1

dx j

∫ n+1∏j=1

dp j

2π hexp

{i

h

n+1∑j=1

[p j (x j − x j−1)

−H(p j , x j−1)(t j − t j−1)]

}(2.6)

2.1 Path integrals in quantum mechanics 73

where the limit n → ∞ (ε → 0) has been taken and the O(ε2)

terms neglected.This result we shall write in the compact form

〈x ′, t ′|x, t〉 =∫ Dx Dp

2π hexp

{i

h

∫ t ′

t[px − H(p, x)] dτ

}(2.7)∫

(Dx Dp/2π h) ≡ ∫ ∏τ [dx(τ ) dp(τ )/2π h] is called a functional integration over

all phase space, with the boundary conditions x(t) = x , x(t ′) = x ′ implied in thiscase. Eq. (2.7) is the promised path integral representation of 〈x ′, t ′|x, t〉.

If the hamiltonian is of a simple form

H = (1/2m)P2 + V (X) (2.8)

it is convenient to perform the momentum integrations in (2.6). Shifting theintegration variables: p j → p j − m( x j/ε) we obtain∫

dp j

2π hexp

[i

h

(p j x j −

p2j

2mε

)]= 1

N j

1

2π hexp

[i

hε

m

2

( x j

ε

)2](2.9)

where x j = x j − x j−1 and

1

N j=∫

dp j exp

(− i

h

p2j

2mε

)The final result has the form of a functional integral over configuration space:

〈x ′, t ′|x, t〉 = 1

N

∫ Dx

2π hexp

{i

hS[x]

}(2.10)

Here S[x] = ∫ t ′t L(x, x) dτ is the action integral over the trajectory x(τ ), where

L(x, x) = 12 mx2 − V (x) is the Lagrange function and the normalization factor N

is given by

1

N=∫

Dp exp

(− i

h

∫ t ′

t

p

2mdτ

)Starting with the canonically quantized theory described by hamiltonian (2.8)

we have derived path integral representation (2.10). We can use another approach,namely, to define the quantum theory by functional integral (2.10), i.e. we canchoose the path integral formulation as the quantization prescription for a systemwith the classical hamiltonian in the form (2.8). Then our derivation proves theequivalence of the path integral and canonical quantization methods for systemsdescribed by hamiltonian (2.8). There are, however, systems for which canonicalquantization is ambiguous due to non-commutativity of operators P and X . Herebelong, for instance, theories where the lagrangian is

L = 12 f (x)x2 + g(x)x − V (x) (2.11)

74 2 Path integrals

On the other hand, quantum theory, which in the classical limit gives the theory(2.11), is unambiguously defined by the path integral ansatz in configuration space,insensitive to the ordering of P and X . It can be shown (Cheng 1972) that theappropriately generalized path integral reads

〈x ′, t ′|x, t〉 ∼ limj→∞

∫ ∏t j

dx(t j )

2π hf 1/2(x(t j )) exp

{i

hS[x(t j )]

}(2.12)

Equivalently, one can postulate (2.7) with the c-number hamiltonian correspondingto lagrangian (2.11), supplemented by the rule that, whenever there is an ambiguity,the integrals over p j are to be performed before the x-integration. Integratingover p j one derives (2.12) (see Problem 2.1). The modification of the functionalmeasure Dx into

Dx = limi→∞

∫ ∏i

dxi f 1/2(xi )

comes from the integral

1

N j=∫

dp j exp

[− i

2hf −1(x j )p2

jε

]which now depends on x j . The functional integral (2.12) defines the same dynam-ics as the Schrodinger equation with one particular ordering of the operators P andX , namely the symmetric ordering, for example, [ f (x)p]S = 1

2 [ f (x)p + p f (x)].Theories like (2.11) are of physical interest. A lagrangian of this kind appears,

for instance, for a system of two interacting particles

L = m

2x2 + m

2y2 − V (x, y) (2.13)

subject to a constraint equation y = c(x). Therefore

L = m

2

[(1 + dc

dx

)2

x2

]− V (x, c(x)) (2.14)

We shall encounter a similar situation in Chapter 3 when quantizing gauge fieldtheories.

In summary, in the following we shall consider quantum theories defined by thepath integral formulation. Their correspondence to canonically quantized theories,if of interest, must be verified case by case. For field theories considered in thisbook this, in fact, can be done at least in the framework of perturbation theory.

If S[x(τ )] � h we can evaluate (2.10) by use of the saddle point approximation.The integral is then dominated by trajectories close to the classical one xc(t), whichsatisfies δS[xc(τ )] = 0: i.e. the path integral formulation allows a relatively simpleunderstanding of the classical and semi-classical limit.

2.1 Path integrals in quantum mechanics 75

Matrix elements of position operators

The matrix element 〈x ′, t ′|x, t〉 determines all transition probabilities betweenquantum mechanical states. In view of further applications of functional formalismto quantum field theories it is also important to know the path integral represen-tation of the matrix elements of the position operators, corresponding to the fieldoperator of quantum field theory. For the time-ordered product of N such operatorsthe following expression can be shown to hold

〈x ′, t ′|T X (t1) . . . X (tN )|x, t〉 =∫ Dx Dp

2π hx(t1) . . . x(tN )

× exp

{i

h

∫ t ′

t[px − H(p, x)] dτ

}(2.15)

Let us check (2.15) for the product of two operators: X (τ1)X (τ2) at τ1 > τ2. Againwe divide the time axis into small intervals, choosing t1 . . . tn in such a way that

τ1 = ti1, τ2 = ti2

and apply the completeness relation at each ti . We obtain

〈x ′, t ′|X (τ1)X (τ2)|x, t〉 =∫ ∏

i

dxi 〈x ′, t ′|xn, tn〉 · · · 〈xi1, ti1 |X (τ1)|xi1−1, ti1−1〉 · · ·

×〈xi2, ti2 |X (τ2)|xi2−1, ti2−1〉 · · · 〈x1, t1|x, t〉=∫ ∏

i

dxi xi1 xi2 〈x ′, t ′|xn, tn〉 · · · 〈x1, t1|x, t〉 (2.16)

Proceeding exactly as when deriving (2.6) we obtain (2.15). Note that (2.16) is truefor τ1 > τ2. When τ1 < τ2, the r.h.s. of (2.16) corresponds to the matrix element〈x ′, t ′|X (τ2)X (τ1)|x, t〉. Therefore the path integral, like (2.15), defines the matrixelement of the time-ordered product of the position operators∫ Dx Dp

2π hx(t1) x(t2) exp

{i

h

∫ t ′

t[px − H ] dτ

}={〈x ′, t ′|X (t1)X (t2)|x, t〉, t1 > t2

〈x ′, t ′|X (t2)X (t1)|x, t〉, t1 < t2(2.17)

As before, it is also possible to change from phase space path integrals to pathintegrals over configuration space.

Also, let us note that the transition amplitude in the presence of an externalsource J (τ )

〈x ′, t ′|x, t〉J =∫ Dx Dp

2π hexp

{i

h

∫ t ′

t[px − H(p, x)+ h J (τ )x(τ )] dτ

}(2.18)

76 2 Path integrals

corresponding to the usual transition amplitude with the hamiltonian modified bya source term: H → H − h J x , can be used as a generating functional of thematrix elements of the position operators, which are then given by its functionalderivatives with respect to J (τ )

〈x ′, t ′|T X (t1) . . . X (tN )|x, t〉 =(

1

i

)NδN

δ J (t1) . . . δ J (tN )

∣∣∣∣J=0

〈x ′, t ′|x, t〉J (2.19)

Instead of a formal definition of the functional derivative, for our purpose it issufficient to know that, in the case of the functional F[J ] defined as

F[J ] =∫

dq1 . . .

∫dqn f (q1 . . . qn) J (q1) . . . J (qn) (2.20)

the functional derivative with respect to J is simply ( f can be chosen to besymmetric in all variables)

δF[J ]

δ J (q)=∫

dq1 . . . dqn−1 J (q1) . . . J (qn−1) n f (q1 . . . qn−1q) (2.21)

This corresponds to the usual rule of differentiating the monomials. If thefunctional is defined instead by the series

�[J ] =∞∑

n=1

1

n!

∫dq1 . . . dqn φn(q1 . . . qn) J (q1) . . . J (qn) (2.22)

then

φn(q1 . . . qn) = δn�[J ]

δ J (q1) . . . δ J (qn)

∣∣∣∣J=0

(2.23)

which compares with the Taylor expansion of the usual functions.

2.2 Vacuum-to-vacuum transitions and the imaginary time formalism

General discussion

In field theoretical applications we shall mainly deal with the matrix elements ofthe field operators taken between the vacuum states: the Green’s functions. Letus first consider the analogous problem in quantum mechanics. Assume that thelagrangian L of the system is time-independent. The energy eigenstates correspondto the wave-functions �n(x) = 〈x |n〉. In particular, the ground state or the vacuumis described by the function �0(x) = 〈x |0〉. It will be convenient to use �0(x, t)defined as

�0(x, t) = exp[−(i/h)E0t]〈x |0〉 = 〈x | exp[−(i/h)Ht]|0〉 = 〈x, t |0〉 (2.24)

2.2 Vacuum-to-vacuum transitions 77

We are interested in the matrix element 〈0|T X (t1) . . . X (tn)|0〉. It reads

〈0|T X (t1) . . . X (tn)|0〉 =∫

dx ′ dx �∗0(x

′, t ′) 〈x ′, t ′|T X (t1) . . . X (tn)|x, t〉�0(x, t)

(2.25)and for the matrix element 〈x ′, t ′|T X (t1) . . . X (tn)|x, t〉 the functional form (2.15)can be used. The considered vacuum expectation value can also be obtained froma generating functional

〈0|T X (t1) . . . X (tN )|0〉 =(

1

i

)NδN

δ J (t1) . . . δ J (tN )

∣∣∣∣J=0

W [J ] (2.26)

where

W [J ] = 〈0|0〉J =∫

dx ′ dx �∗0(x

′, t ′) 〈x ′, t ′|x, t〉J�0(x, t) (2.27)

with 〈x ′, t ′|x, t〉J given by (2.18). It is very important that the generating functionalW [J ] can also be derived in another way. We shall show that

W [J ] = limT1→+i∞T2→−i∞

exp[(i/h)E0(T2 − T1)]

�∗0(x1)�0(x2)

〈x2, T2|x1, T1〉J (2.28)

This implies that W [J ] is, in fact, determined by the transition amplitude〈x2, T2|x1, T1〉J at any given x1, x2 (for instance x1 = x2 = 0) provided that theanalytic continuation to the imaginary values of T1 and T2 is performed. To derive(2.28) let us choose that the source J (t) vanishes outside the time interval (t, t ′)with T2 > t ′ > t > T1. Then we can write

〈x2, T2|x1, T1〉J =∫

dx ′ dx 〈x2, T2|x ′, t ′〉〈x ′, t ′|x, t〉J 〈x, t |x1, T1〉 (2.29)

where

〈x, t |x1, T1〉 = 〈x | exp[−(i/h)H(t − T1)]|x1〉=∑

n

�n(x)�∗n(x1) exp[−(i/h)En(t − T1)]

and similarly for 〈x2, T2|x ′, t ′〉. The only T -dependent terms are now the factorsexp[−(i/h)En(t − T1)] and we can continue to T1 → +i∞ explicitly. Recallingthat E0 is the lowest energy eigenvalue we get

limT1→+i∞

exp[−(i/h)E0T1]〈x, t |x1, T1〉 = �0(x) exp[−(i/h)E0t]�∗0(x1)

= �0(x, t)�∗0(x1)

and, in the same way

limT2→−i∞

exp[(i/h)E0T2]〈x2, T2|x ′, t ′〉 = �∗0(x

′, t ′)�0(x2)

78 2 Path integrals

After employing (2.27) and (2.29), (2.28) follows. Notice also that (2.28) and(2.26) imply

〈0|T X (t1) . . . X (tN )|0〉 = limT1→+i∞T2→−i∞

exp[(i/h)E0(T2 − T1)]

�∗0(x1)�0(x2)

×〈x2, T2|T X (t1) . . . X (tN )|x1, T1〉 (2.30)

We conclude that the vacuum matrix elements can be calculated by takingfunctional derivatives of the generating functional W [J ], given by (2.28). TheJ -independent factors are irrelevant, because we can always consider quantitieslike

1

〈0|0〉〈0|T X (t1) . . . X (tN )|0〉 =(

1

i

)N 1

W [0]

δN

δ J (t1) . . . δ J (tN )

∣∣∣∣J≡0

W [J ] (2.31)

Consequently, instead of (2.28) we can write

W [J ] = limT1→+i∞T2→−i∞

∫x(T1)=x1x(T2)=x2

Dx exp

{(i/h)

∫ T2

T1

[L(x, x)+ h J x] dt

}(2.32)

where x1, x2, are arbitrary.

Harmonic oscillator

As an example of particular interest in view of further field theoretical applicationswe shall consider the case of the harmonic oscillator

L = 12 x2 − 1

2ω2x2 = 1

2(d/dt)(x x)− 12 x x − 1

2ω2x2

The action integral, including the source term, can be written as follows

S J [x] =∫ T2

T1

[L + h J x] dt = 12 x x

∣∣T2

T1+ 1

2(x, Ax)+ (h J, x) (2.33)

where ( f, g) denotes the ‘scalar product’

( f, g) =∫ T2

T1

dt f (t) g(t)

and the operator A is defined by the equation∫ T2

T1

dt dt ′ x(t) A(t, t ′) x(t ′) =∫ T2

T1

(−x x − ω2x2)

dt (2.34)

so that

A(t, t ′) = −(d2/

dt ′ 2 + ω2)δ(t ′ − t) (2.35)


and

Ax ≡∫ T2

T1

dt ′ A(t, t ′) x(t ′)

Let us now change the functional integration variable in (2.32) by introducingz(t)

x(t) = xc(t)+ z(t)

where xc(t) is a solution of the classical equation of motion (in the presence of thesource)

x + ω2x = h J

Requiring that xc(t) satisfies the boundary conditions xc(T1) = x1, xc(T2) = x2,we have z(T1) = z(T2) = 0. The classical trajectory xc(t) can be written as asuperposition of a solution x0(t) of the homogeneous equation, and the specialsolution of the complete equation

xc(t) = x0(t)+ h∫

dt ′ G(t − t ′) J (t ′) (2.36)

Here G(t − t ′) is the Green’s function of the classical equation of motion(d2/

dt2 + ω2)G(t − t ′) = δ(t − t ′) (2.37)

so that G = −A−1.† Re-expressing the action integral (2.33) in terms of theclassical trajectory and the new integration variable z(t) we get

S J [xc + z] = 12 xc xc

∣∣T2

T1+ 1

2(z, Az)+ 12 h(J, x0)+ 1

2 h2(J, G J ) (2.38)

where we have used the relations

Axc = −h J, (xc, Az) = (Axc, z)− zxc

∣∣T2

T1

together with (2.36).The resulting W [J ] is of the following form

W [J ] = limT1→+i∞T2→−i∞

exp[(i/2h)h2(J, G J )

]exp

{(i/2h)

[xc xc

∣∣T2

T1+ h(J, x0)

]}×∫

z(T1)=0z(T2)=0

Dz exp

[(i/h)

∫ T2

T1

L(z) dt

](2.39)

The important observation is that if the Green’s function (2.37) is chosen tosatisfy the conditions

G(t) −→t→±i∞

0, dG(t)/dt −→t→±i∞

0 (2.40)

† In the space of functions x(t) with zero modes x0(t) of A excluded.

80 2 Path integrals

the only functional dependence of W [J ] on J is given by the factor

W [J ] ∼ exp[(i/2h)h2(J, G J )

](2.41)

To prove this assertion we should show that xc xc

∣∣T2

T1and (J, x0) are J -independent

whenever the boundary conditions (2.40) are satisfied. In fact (2.36) and (2.40)imply

xc xc

∣∣T2

T1−→

T1→+i∞T2→−i∞

x0 x0

∣∣T2

T1

with x0 obviously J -independent. For (J, x0) the argument is somewhat moreinvolved. Consider the solution of the homogeneous equation

x0(t) = A exp(iωt)+ B exp(−iωt)

with A, B chosen to satisfy the boundary conditions x0(T1) = x1, x0(T2) = x2

A = x2 exp(−iωT1)− x1 exp(−iωT2)

exp[iω(T2 − T1)] − exp[−iω(T2 − T1)],

B = x1 exp(iωT2)− x2 exp(iωT1)

exp[iω(T2 − T1)] − exp[−iω(T2 − T1)]

In the limit T1 →+i∞, T2 →−i∞ we have

A ≈ x2 exp(−ω|T2|), B ≈ x1 exp(−ω|T1|)implying that (J, x0) also vanishes in this limit.

Note that in (2.41) the limit T1 → +i∞, T2 → −i∞ has in fact been omitted.This could be done because the integral∫ T2

T1

dt∫ T2

T1

dt ′ J (t) G(t − t ′) J (t ′) ≡ (J, G J )

is independent of the limits of integration due to the requirement that J (t)vanishes outside some finite time interval (t1, t2) such that T2 > t2 > t1 > T1.The dependence on T1 and T2 can now occur only in the factors which areJ -independent because of the boundary conditions (2.40), and therefore irrelevantfor our discussion.

To complete the case of the harmonic oscillator we must find the Green’sfunction G(t − t ′) consistent with the boundary conditions specified by (2.40).Introducing the Fourier transform G(ν)

G(t) =∫ ∞

−∞

dν

2πG(ν) exp(−iνt)


Fig. 2.1. Possible prescriptions for bypassing the singularity at ν = ±ω in the integral(2.42) in the complex ν plane.

we obtain the formal solution of (2.37)

G(t) = −∫ ∞

−∞

dν

2π

1

ν2 − ω2exp(−iνt) (2.42)

This result represents, in fact, four distinct solutions depending on the prescriptionfor bypassing the singularity at ν = ±ω, which may be one of those shown inFig. 2.1. Cases (a) and (b) are easily seen to be in conflict with the boundaryconditions (2.40). It suffices to perform the analytic continuation of (2.42) bydeforming the contour as shown in the picture and then to integrate for imaginaryt . The two remaining solutions agree with (2.40): we again continue analyticallyto imaginary t by deforming the contour of the ν-integration from the real to theimaginary axis. They correspond to adding to the denominator of the integrand in(2.42) (+iε) and (−iε), respectively

G(t) = −∫ ∞

−∞

dν

2π

1

ν2 − ω2 ± iεexp(−iνt) (2.43)

Adding (+iε) to the Green’s function denominator is equivalent to introducing theextra term

(±iεx2)

to the lagrangian function. Only for(+iεx2

)is the functional

integral well defined, so that the final solution for the Green’s function is specifieduniquely

G(t) = −∫ ∞

−∞

dν

2π

1

ν2 − ω2 + iεexp(−iνt) (2.44)

After the contour integration this becomes (for real t)

G(t) = i

2ω[exp(−iωt)�(t)+ exp(iωt)�(−t)] (2.45)

The physical interpretation of the (+iε) prescription is thus made clear: itcorresponds to the positive frequency solutions propagating into the future.

82 2 Path integrals

Fig. 2.2. Rotation of the integration contour from the real to the imaginary axis in thecomplex ν-plane which defines the analytic continuation of GE(τ ) to τ = it .

Euclidean Green’s functions

Now let us observe that the same Green’s function can also be derived in anotherway, i.e. by the analytic continuation from the imaginary time region. To this endwe first define the ‘Euclidean’ (i.e. imaginary time) Green’s function GE(τ ), whichobeys the equation [−(d2

/dτ 2

)+ ω2]GE(τ ) = δ(τ ) (2.46)

Consequently,

GE(τ ) =∫ ∞

−∞

dν

2π

1

ν2 + ω2exp(−iντ) (2.47)

As the singularities of the integrand appear at ν = ±iω, away from the integrationcontour, GE(τ ) is uniquely specified to be

GE(τ ) = −(

1

2ω

)exp(−ω|τ |) (2.48)

in agreement with the boundary conditions specified for G(t). We shall now showthat the real time Green’s function G(t) is given by the analytic continuation ofGE(τ ) to τ = it

G(t) = −iGE(it) (2.49)

In order to perform this continuation the integration contour in (2.47) should bedeformed to keep the integral convergent during the τ = it ; such a rotation isillustrated in Fig. 2.2. The result is

G(t) = −∫ ∞

−∞

dν

2π

1

ν2 − ω2 + iεexp(−iνt)

which coincides with (2.44).It is also possible to continue to τ = −it . In this case, however, the resulting real

2.3 Path integral formulation of quantum field theory 83

time Green’s function appears with the opposite, (−iε), prescription. We concludethat the analytic continuation is correct for τ = it . One way to see the difference isto make the substitution t = ∓iτ under the functional integral. The action changesas follows:

i

hS[x] = i

h

∫dt

[1

2

(dx

dt

)2

− V (x)

]−→t=±iτ

i

h(±i)

∫dτ

[−1

2

(dx

dt

)2

− V (x)

]= ±1

hSE[x] (2.50)

where SE[x] is the ‘Euclidean’ action. The resulting functional integral is of theform ∫

Dx exp{±(1/h)SE[x]} (2.51)

and only for the (−) sign is it well defined.For the generating functional WE[J ] one gets

WE[J ] =∫

Dx exp

{−(1/h)SE[x] +

∫dτ J (τ ) x(τ )

}∼ exp

[12 h(J, GE J )

]where GE(τ ) is defined by (2.46).

2.3 Path integral formulation of quantum field theory

Green’s functions as path integrals

The results of Sections 2.1 and 2.2 are easily generalized to the case of morethan one degree of freedom. If the number of degrees of freedom is to be n,the coordinate x should be replaced by an n-component vector. The functionalintegral would now correspond to the sum over all trajectories in the n-dimensionalconfiguration space, satisfying appropriate boundary conditions.

In the field theory, the trajectory x(t) is replaced by a field function �(x, t).The degrees of freedom are now labelled by the continuous index x; the number ofdegrees of freedom is obviously infinite. To define the appropriate path integral onecan start from a multiple integral on a discrete and, for a beginning, finite lattice ofspace-time points. This amounts to defining the quantum field theory as a limit ofa theory with only a finite number of degrees of freedom.

The limit of an infinite lattice, related to the thermodynamical limit of statisticalmechanics, already defines a theory with an infinite number of degrees of freedom.However, this lattice theory does not have enough space-time invariance and acontinuous theory must be defined. The latter limit is accompanied by infinities,the ‘UV divergences’ of quantum field theory. The definition of the functionalintegral in quantum field theory is thus more ambiguous than in the case of quantum

84 2 Path integrals

mechanics. Nevertheless, the functional formalism in quantum field theory is ofgreat heuristic value. It is a very convenient tool for studying perturbation theoryand allows a natural description of some non-perturbative phenomena.

The quantum field theory is usually formulated in terms of the vacuum expec-tation values of the chronologically ordered products of the field operators, theGreen’s functions:

G(n)(x1, . . . , xn) = 〈0|T�(x1) . . . �(xn)|0〉 (2.52)

Using our experience from the previous section we shall write down the pathintegral representation for them. In particular, it is important to remember the roleplayed by (2.28) and (2.30) in getting rid of the vacuum wave functions originallypresent in (2.25) as the boundary conditions. Thus, by analogy with the results ofthe previous section, we postulate the following path integral representation

G(n)(x1, . . . , xn) ∼∫

D��(x1) . . . �(xn) exp

[(i/h)

∫d4x L

](2.53)

(D� denotes integration over all functions �(x, t) of space and time, because, foreach value of x, �(x, t) corresponds to a separate degree of freedom; L is thelagrangian density). Eq. (2.30) suggests that any boundary conditions are, in fact,irrelevant here, provided that we take the imaginary time limit. In particular, thetrajectories may be not constrained at all.

In analogy with (2.30), the rotation of the time contour away from the real axisshould start somewhere at large absolute values of time, so that the arguments of theGreen’s functions, t1 . . . tn , stay real. If instead the whole time axis is rotated, ti =−iτi , the result is a Euclidean Green’s function (see the end of Section 2.2). Thelatter has a particularly convenient path integral representation, because the weightfactor in the integrand, exp(−SE/h), is then non-negative. This Euclidean pathintegral formalism can be used to define the Minkowski space Green’s functionsby an analytic continuation of the Euclidean ones. An expression like (2.53) wouldthen be understood as an analytic continuation in the variables t1, . . . , tn of theanalogous Euclidean formula. Equivalently one can work in Minkowski space anduse the (+iε) prescription (see (2.62)).

Eq. (2.53) should be regarded as the formulation of the theory. What is therelation between the path integral and the usual (canonical) operator formulation ofquantum field theory based on the same lagrangian L? For our purposes these areequivalent if they imply the same perturbation theory Feynman rules. This has to bechecked in each case of interest. However, the derivation of the perturbation theoryrules is much simpler in the functional framework, particularly in the case of gaugefield theories. In the following we shall often use the path integral formulation,


referring to the operator formalism only if it helps to make the presentation moreconcise, like in the case of the scattering operator discussed in Section 2.7.

It is convenient to normalize the Green’s functions by factorizing out the vacuumamplitude

G(n)(x1, . . . , xn) = 〈0|T�(x1) . . . �(xn)|0〉/〈0|0〉= N

∫D��(x1) . . . �(xn) exp

[(i/h)

∫d4x L

](2.54)

where

1/N =∫

D� exp

[(i/h)

∫d4x L

]∼ 〈0|0〉 (2.55)

This removes the extra factors like those appearing in (2.30). The T -productdefined by (2.54) is a covariant quantity (see Section 10.1 on ambiguities ofT -products). The derivatives of the Green’s functions are functional integrals ofderivatives of fields, e.g.

�x G(2)(x, y) ∼∫

D��x �(x)�(y) exp

[(i/h)

∫d4x L

](2.56)

The Green’s functions (2.54) satisfy the standard equations of motion which canbe derived using the invariance of the functional integral under a shift of variables�(x) → �(x) + ε f (x) (see Problem 2.2): for example, in a scalar massless fieldtheory defined by the lagrangian

L = 12∂µ�∂µ�− V (�)

we have

�x G(2)(x, y) = −〈0|T V ′(�(x))�(y)|0〉 − iδ(x − y) (2.56a)

We observe also that given a theory formulated in terms of the Green’s functions(2.54) quantities like 〈0|T �x�(x)�(y)|0〉 remain a priori undefined. We candefine them by requiring that they satisfy certain constraints like, for example,for a scalar massless field theory

�x〈0|T�(x)�(y)|0〉 = 〈0|T �x�(x)�(y)|0〉 − iδ(x − y)

where the l.h.s. is given by (2.56a), which amounts to specifying a regularizationprescription for them.

The Green’s functions (2.54) are given by the functional derivatives of thefunctional W [J ] equivalent to the vacuum transition amplitude in the presence ofthe external source J (x)

W [J ] = N∫

D� exp

{(i/h)

∫d4x [L+ h J (x)�(x)]

}(2.57)

86 2 Path integrals

Expanding in powers of J and using (2.54) we can rewrite W [J ] as follows:

W [J ] =∞∑

n=0

in

n!

∫dx1 . . . dxn G(n)(x1, . . . , xn) J (x1) . . . J (xn)

Consequently,

G(n)(x1, . . . , xn) =(

1

i

)nδn

δ J (x1) . . . δ J (xn)

∣∣∣∣J≡0

W [J ] (2.58)

The Green’s functions can also be considered as the analytic continuation ofthose obtained from the generating functional defined in the Euclidean space withx0 = −ix4, where x4 is real:

WE[J ] = N∫

D� exp

{−(1/h)SE[�(x)] +

∫d3x dx4 J�

}where, for example, for a free scalar field theory

SE[�(x)] = 12

∫d3x dx4

[(∂�

∂ x4

)2

+ (∇∇∇�)2 + m2�2

].

The Euclidean Green’s functions are given by

G(n)E (x1, . . . , xn) = δ

δ J (x1). . .

δ

δ J (xn)

∣∣∣∣J≡0

WE[J ]

Using path integral formalism one can derive equations of motion for them(Problem 2.2): for example, in free scalar field theory(

− ∂2

∂ x24

−∇∇∇2 + m2

)G(2)

E (x, y) = δ(x4 − y4)δ(x − y)

analogous to those in Minkowski space.The Minkowski space Green’s functions are given by analytic continuation

G(n)(x1, . . . , xn) = (i)nG(n)E (x1, . . . , xn)

where

x = x, x0 = −ix4

Since in Euclidean space the exponent in the generating functional WE[J ] isbounded from above we can evaluate the functional integral by the saddle pointmethod (see Problem 2.8). We recall that the saddle point approximation rests onthe fact that the integral

I =∫

dx exp[−a(x)]


can be successfully approximated by

I ∼= exp[−a(x0)]∫

dx exp[− 1

2(x − x0)2a′′(x0)

]where x0 satisfies a′(x0) = 0, if a′′(x0) > 0 and if the points away from theminimum do not contribute much.

We remark also that we can define the field operator evolution for imaginarytime

�(x) = exp(iH x0)�(0, x) exp(−iH x0) = exp(H x4)�(0, x) exp(−H x4)

and also the operator ordering T with respect to x4 analogously to the T -productdefinition, for example,

T�(x)�(y) = �(x4 − y4)�(x)�(y)+�(y4 − x4)�(y)�(x)

whose vacuum expectation value then satisfies the same equation of motion asG(2)

E (x, y) defined by the generating functional in Euclidean space.

Action quadratic in fields

In the case in which the classical action is quadratic in the field variable �(x)

S = 12

∫d4x d4 y �(x) A(x, y)�(y) (2.59)

the generating functional W0[J ] (index 0 corresponds to the specific form (2.59)of the action) is easily obtained in a closed form. Repeating the steps that led us tothe derivation of formula (2.41) we obtain

W0[J ] = exp

[(i

2h

)h2∫

d4x d4 y J (x) G(x, y) J (y)

](2.60)

where G(x, y) is the Green’s function of the classical field equation∫A(x, y)�(y) d4 y = −h J (x) (2.61)

satisfying conditions (2.40).One example of a theory with quadratic action is the theory of a free field, with

the lagrangian density

L = 12

(∂µ�∂µ�− m2�2

)+ 12 iε�2 (2.62)

The extra term 12 iε�2 has been added in accordance with the discussion following

88 2 Path integrals

(2.43): it makes the functional integral (2.57) well-defined and simultaneouslyensures the correct boundary conditions for the Green’s function G(x, y). As

∂µ�∂µ� = ∂µ(�∂µ�

)−�∂µ∂µ� (2.63)

and neglecting in the action the surface term at infinity, the action can be written inthe form

S = 12

∫d4x d4 y �(x)

[−∂µ∂µ − m2 + iε

]δ(x − y)�(y) (2.64)

i.e.

A(x, y) = −[∂µ∂µ + m2 − iε]δ(x − y) (2.65)

The classical field equation(∂µ∂

µ + m2 − iε)�(x) = h J (x) (2.66)

is then solved by

�(x) = h∫

d4 y G(x − y) J (y) (2.67)

with the Green’s function G(x − y) satisfying[∂µ∂

µ

(x) + m2 − iε]G(x − y) = δ(x − y) (2.68)

Introducing the Fourier transform G(k)

G(x − y) = 1

(2π)4

∫d4k G(k) exp[−ik(x − y)] (2.69)

we obtain from (2.68)

G(k) = − 1

k2 − m2 + iε(2.70)

and therefore

G(x − y) = 1

(2π)4

∫d4k

1

k2 − m2 + iεexp[−ik(x − y)] (2.71)

Again, the Green’s function agrees with the analytic continuation from the Eu-clidean region.

Gaussian integration

We end this section with a comment on gaussian integration. Having definedfunctional integrals as an appropriate limit of multiple integrals, one encounters


integrals of the type

I =∫ ∞

−∞dx exp

(−bεx2 + ibx2) = 2

∫ ∞

0dx exp

[−bx2(ε − i)]

(2.72)

with ε → 0, where the ε-term corresponds to the (+iε) prescription. Rotating thecontour of integration into x ′ = x exp(iϕ) such that exp(2iϕ)(ε− i) = 1 one easilyfinds

I = exp[i(

14π − 1

2ε)](π/b)1/2 −→

ε→0(iπ/b)1/2 (2.73)

which is the analytic continuation of the gaussian integral∫ ∞

−∞dx exp

(−ax2) = (π/a)1/2 (2.74)

for complex a (and Re a > 0). A similar result holds for integration over thecomplex variable z = x + iy∫

dz∗ dz exp(−az∗z) ≡ 2∫ ∞

−∞dx∫ ∞

−∞dy exp

[−a(x2 + y2

)] = 2π/a (2.75)

for complex a (and Re a > 0). We write∫

dz∗ dz for 2∫

d Re z∫

d Im z byconvention. For many degrees of freedom z1, . . . , zn we can define the complexvector z

z =

z1...

zn

and denote by (z∗, Az) the scalar product of vectors z and Az, where A is a(n × n)-dimensional complex matrix. Then we have the following generalizationof relation (2.75):∫

dz∗1 dz1 . . .

∫dz∗n dzn exp[i(z∗, Az)] = (2π)nin/ det A (2.76)

for any positive-definite matrix A which can be diagonalized by unitary transfor-mations. The result is obvious for a diagonal matrix A (it follows from multipleuse of (2.75)). For the general case one has to show that the integration measureis invariant under the unitary transformation diagonalizing A (the scalar product(z∗, Az) is obviously invariant). Denote the unitary transformation by U = A+iB,with A and B real matrices. From the unitarity of U : UU † = 1 it follows that

AAT + B BT = 1, B AT − ABT = 0 (2.77)

90 2 Path integrals

The change of integration variables z′ = U z can be written as(x ′

y′

)=(

A −BB A

)(xy

)= M

(xy

)(2.78)

and using (2.77) we conclude that the transformation matrix M is orthogonal(M MT = 1). Therefore the integration measure is indeed invariant under theunitary transformation U . For integration over n real variables x ≡ (x1, . . . , xn),one has ∫

dx1 . . . dxn exp[i(x, Ax)] = πn/2in/2/[det A]1/2 (2.79)

In the limit n → ∞ integral (2.79) is the J -independent path integral in (2.39).Although we do not have to evaluate it explicitly, it is worth remembering itscompact form (2.79). Another point is that in addition to results (2.76) and (2.79)we can also easily calculate integrals in which the product (z∗, Az) is replaced byan expression

(z∗, Az)+ (b∗, z)+ (z∗, b) ≡ F(z) (2.80)

where b is a constant vector, and matrix A is now hermitean. Indeed, expression(2.80) can be written as

F(z) = (ω∗, Aω)− (b∗, A−1b

)(2.81)

where ω = z − z0 and z0 = −A−1b is the minimum of F(z). Actually, (2.81)has been used in the derivation of (2.41) and (2.60) for the generating functionalW0[J ].

Finally, we recall that for any matrix L which can be diagonalized by a unitarytransformation

det(1 − L) = exp Tr ln(1 − L)

where

Tr ln(1 − L) = −Tr[L + 1

2 L2 + 13 L3 + · · · ]

2.4 Introduction to perturbation theory

Perturbation theory and the generating functional

We shall discuss first a simple case of a scalar field theory described by thelagrangian

L = 12

[∂µ�(x)∂µ�(x)− m2�2(x)

]− λ

4!�4(x) (2.82)

where x denotes four coordinates in Minkowski space. We seek a method forcalculating an arbitrary Green’s function in such a theory. For that purpose it is

2.4 Introduction to perturbation theory 91

convenient to use the functional formulation of the theory in which the Green’sfunctions are given by functional derivatives of the generating functional W [J ]

G(n)(x1, . . . , xn) =(

1

i

)nδ

δ J (x1). . .

δ

δ J (xn)

∣∣∣∣J≡0

W [J ] (2.83)

where

W [J ] = N∫

D� exp

{(i/h)

∫d4x [L+ h J (x)�(x)]

}(2.84)

and factor N is defined by (2.55).We recall at this point that, so far, we have been able to calculate exactly the

generating functional W0[J ] for a theory of non-interacting scalar fields with theaction S0 given by

S0 =∫

d4x Lfree =∫

d4x 12

(∂µ�∂µ�− m2�2

)(2.85)

or, more generally, containing terms at most quadratic in fields. The analogous,exact, solution is not known for the full theory (2.82) and the method to be usedfor calculating the Green’s functions (2.83) is a perturbative expansion in terms ofpowers of SI defined as follows

SI = S − S0 (2.86)

where

S =∫

d4x L

Such an expansion should be understood as an expansion in a neighbourhood ofthe vacuum state for which �(x) = 0, so that the action SI can be regarded as asmall parameter. It is convenient to note the following functional identity∫

D� exp

(i

h

{S0[�] + SI[�] + h

∫J (x)�(x) d4x

})= exp

{i

hSI

[1

i

δ

δ J

]} ∫D� exp

(i

h

{S0[�] + h

∫J (x)�(x) d4x

})(2.87)

so that (2.84) can be rewritten as

W [J ] =exp

{i

hSI

[1

i

δ

δ J

]}W0[J ]

exp

{i

hSI

[1

i

δ

δ J

]}W0[J ]

∣∣∣∣J≡0

= N exp

{i

hSI

[1

i

δ

δ J

]}W0[J ] (2.88)

The perturbation series is generated by expanding the exponential factor

92 2 Path integrals

exp{(i/h) SI[(1/i)(δ/δ J )]} in powers of SI and performing the functional differ-entiations as indicated. This is equivalent to expanding exp{(i/h)SI[�]} under thepath integral. In perturbation theory one gets, therefore, the following generalformula for the Green’s functions (2.83)

G(n)(x1, . . . , xn) =

∫D��(x1) . . . �(xn)

[ ∞∑N=0

1

N !

(i

hSI

)N]exp

(i

hS0

)∫

D�

[ ∞∑N=0

1

N !

(i

hSI

)N]exp

(i

hS0

)(2.89)

If SI can be written as an integral over polynomials in fields (as in our example)any Green’s function can be calculated term by term using (2.89). Each term canbe written as a respective derivative of the functional W0[J ].

Wick’s theorem

As an introduction to the detailed exposition of perturbative rules we consider againthe case of non-interacting scalar fields and prove Wick’s theorem. The n-pointGreen’s function G(n)

0 (x1, . . . , xn) is given by (2.83) with W [J ] replaced by W0[J ].From the previous considerations we know that

W0[J ] = exp

[12 ih

∫d4x d4 y J (x) G(x − y) J (y)

](2.90)

up to the overall J -independent normalization factor which cancels out in thedefinition of the Green’s functions, where G(x − y) is the Green’s function forthe free classical Klein–Gordon equation

G(x − y) = − 1

(2π)4

∫d4k

exp[−ik(x − y)]

k2 − m2 + iε(2.91)

Referring again to (2.83) one observes that the n-point Green’s function is given bythe

(12 n)th term of the Taylor expansion of exponent (2.90). For the free propagator

one immediately gets

G(2)0 (x1, x2) = −ihG(x1 − x2) (2.92)

(the symmetry G(x1 − x2) = G(x2 − x1) has been taken into account). The casen = 4 is somewhat more complicated. (Notice that all G(n)-functions with n oddequal zero.) It is convenient to convert the integrals in (2.90) into sums over discretepoints and write

i4G(4)0 (x1, . . . , x4) = δ4

δ J1δ J2δ J3δ J4

∣∣∣∣J=0

(i

2h

)2 1

2!

∑i, j,k,l

Ji J j Jk Jl Gi j Gkl (2.93)


where

Ji = J (xi ), Gi j = G(xi − y j )

The structure of (2.93) can be represented diagrammatically by the ‘prototype’diagram

≡ (−ihGi j )× (−ihGkl)

and the differentiation with respect to J1, . . . , J4 leads to all possible assignmentsof subscripts 1–4 to subscripts i, j, k, l. One gets

So finally

G(4)0 (x1, . . . , x4) = G(2)

0 (x1, x2)G(2)0 (x3, x4)+ G(2)

0 (x1, x3)G(2)0 (x2, x4)

+G(2)0 (x1, x4)G

(2)0 (x2, x3) (2.94)

This is the content of Wick’s theorem (see, for example, Bjorken & Drell (1965))which states that any n-point free Green’s function can be written as a sum over allpossible products of 1

2 n two-point Green’s function G(2)0 (xi , x j ).

An example: four-point Green’s function in λ�4

Coming back to our main problem we shall now discuss several examples in orderto derive general rules for perturbative calculations in the theory of interactingfields. Let us consider the four-point Green’s function and discuss the numeratorof (2.89) in the first order in SI which reads∫

D��(x1) . . . �(x4) (−iλ)1

4!h

∫d4 y �4(y) exp

(i

hS0

)(2.95)

94 2 Path integrals

Fig. 2.3.

or, using the generating functional W0[J ],

(−iλ)1

4!h

∫d4 y

δ

δ J (x1)· · · δ

δ J (x4)

δ4

δ J (y)

∣∣∣∣J≡0

W0[J ] (2.96)

It is clear that the non-zero contribution to (2.96) comes from the term in the Taylorexpansion of (2.90) which contains four propagators

(12 ih

)4 1

4!

[ ∫d4x d4 y J (x) G(x − y) J (y)

]4

(2.97)

and can be represented graphically by the prototype diagram in Fig. 2.3. Tomake the combinatorics transparent we have again replaced the integrations in(2.97) by discrete sums. The differentiations of (2.97) now provide all possibleassignments of points x1, x2, x3, x4 and four points y (it is convenient to splitthem into y1, y2, y3, y4, remembering that the limit y1 = y2 = y3 = y4 shouldbe understood) to the points i, j, k, l,m, n, o, p. It is easy to discuss them in asystematic fashion. Let us first assume that pairs of points joined by the propagatorsare specified, for example, (x1, y1), (x2, y2), (x3, y3), (x4, y4). Then, we havean additional combinatorial factor following from the ‘horizontal’ symmetry (forexample, x1 = i , y1 = j and x1 = j , y1 = i) which is 2n for n pairs, andanother factor from the ‘vertical’ symmetry ((x1, y1) can be assigned to (i, j) or(k, l) etc.) which is n! (again for n pairs). These factors cancel with the factor1/(2nn!) present in (2.97) and this happens for any term of expansion (2.89). Sowe have to remember about 1/N ! present in (2.89) (N = 1 in our examples)and about 1/4! present in the definition of the coupling constant and, on the otherhand, we have to take account of all possible choices of pairs of points xi andyi joined by propagators. The result is as shown in Fig. 2.4 with self-evidentclassification of different possibilities. In the limit y1 = y2 = y3 = y4 = ywe get the diagrams shown in Fig. 2.5 where the dot represents the integral of aproduct of two propagators taken at the same space-time points y (notice that thecombinatorial factors in front of the second and the third diagrams include different


Fig. 2.4.

Fig. 2.5.

permutations of x1, x2, x3, x4). The analytic expression for the first diagram reads

(−iλ/h)∫

d4 y G(2)0 (x1, y) G(2)

0 (x2, y) G(2)0 (x3, y) G(2)

0 (x4, y) (2.98)

It remains to extend our considerations to terms SNI , N > 1 and to discuss the role

of the denominator in (2.89). These problems are related since G(n)N involves in

principle a quotient of terms of arbitrary order. We observe that, for example, the S2I

term in the numerator is given by the diagrams (a)–( j) in Fig. 2.6. In general, onecan convince oneself that the numerator of (2.89) can be represented by a product oftwo infinite series of amplitudes as shown in Fig. 2.7. The second bracket containsthe sum of the so-called vacuum amplitudes which do not depend on the Green’sfunction considered and, most importantly, it happens to be just the denominatorof (2.89). (The proof of this again requires some attention to the combinatorialfactors.) So we may ignore the denominator of (2.89) and simultaneously all thediagrams containing vacuum subgraphs. Actually, for a given Green’s functionwe shall be interested only in connected diagrams like those in Fig. 2.8 whichgive non-trivial S-matrix elements for G(4)(x1, . . . , x4). They define the connectedGreen’s functions G(n)

conn. The amplitude of, for example, the central diagram can beeasily read off from the numerator of (2.89). In the following we shall understandthe central diagram to be the one in which the pairs of external points, which arejoined by propagators to each of the interaction points, have been specified. Inmomentum space this corresponds to assigning to each vertex definite externalmomenta. In this sense we have three different central diagrams, each of whichcan be discussed as follows. There are twelve fields under the functional integral,

96 2 Path integrals

Fig. 2.6.

Fig. 2.7.

so the ‘prototype’ diagram with six propagators is relevant for combinatorics. Notwo external points can be joined by a propagator. Two external points must bejoined with the interaction point at y and the other two with the interaction pointat z. In addition two propagators must connect y and z. We can have therefore thesituation depicted in Fig. 2.9 and, in addition, y1 can be replaced by any yi (factor4), y2 by any y j �= yi (factor 3), the same for z4 and z3 (factor 4 × 3), there aretwo combinations for the remaining yl joining zk (factor 2) and finally y can beinterchanged with z in connections with external points (factor 2). We get 4! × 4!to cancel 1/4!× 4! from the definition of the coupling constant. (The factors fromthe ‘vertical’ and ‘horizontal’ symmetries cancel as usual.) Since N = 2 there is


Fig. 2.8.

Fig. 2.9.

an extra factor 12 (we will call it the symmetry factor) and the amplitude reads

12(−iλ/h)2

∫d4 y d4z G(2)

0 (x1, y) G(2)0 (x2, y) G(2)

0 (y, z)

×G(2)0 (y, z) G(2)

0 (x3, z) G(2)0 (x4, z) (2.99)

Our examples can be generalized to the following rules. To calculateG(n)

conn(x1, . . . , xn) to order N in SI draw all connected diagrams consistent with thestructure of the numerator in (2.89). For each diagram calculate the combinatorial(symmetry) factor. Attach (−iλ/h) to each vertex and G(2)

0 to each line.

Momentum space

One is usually interested in Green’s functions in momentum space defined asFourier transforms of the respective Green’s functions in configuration space. Wedefine G(n)(p1, . . . , pn) by the relation

G(n)(x1, . . . , xn) =∫

d4 p1

(2π)4· · · d4 pn

(2π)4exp(−ip1x1) · · · exp(−ipnxn)

×G(n)(p1, . . . , pn) (2.100)

so that all momenta are treated as outgoing (due to (+iε)) in propagators, positivefrequencies propagate into future x0

n → +∞ (see (2.45)). In actual physicaltransitions

x01 . . . x0

k →−∞x0

k+1 . . . x0n →+∞

98 2 Path integrals

Fig. 2.10.

Fig. 2.11.

and therefore the momenta p1, . . . , pn defined by (2.100) are related to the physicalmomenta pi ( p0

i > 0) of the initial and final particles as follows

pi = − pi i = 1, . . . , k

pi = pi i = k + 1, . . . , n

In momentum space we get from (2.92) and (2.91)

G(2)0 (p1, p2) = (2π)4δ(p1 + p2)h

i

p21 − m2 + iε

(2.101)

For the diagram shown in Fig. 2.10, represented by expression (2.98) the momen-tum space amplitude reads

(2π)4δ

( 4∑i=1

pi

)(−iλ/h)

4∏i=1

hi

p2i − m2 + iε

(2.102)

and for the diagram shown in Fig. 2.11 we get from (2.99)

12(−iλ/h)2

4∏i=1

hi

p2i − m2 + iε

∫d4k1

(2π)4

d4k2

(2π)4(2π)4δ(p1 + p2 + k1 + k2)

×(2π)4δ(p3 + p4 − k1 − k2)hi

k21 − m2 + iε

hi

k22 − m2 + iε

(2.103)

The Feynman rules in momentum space are now obvious. Usually, the diagramsas above are assumed to represent amplitudes without the propagators on externallines (Feynman diagrams). Observe that each vertex contributes a factor h−1 and


Fig. 2.12.

each internal line gives a factor h. Thus a Feynman diagram with L loops is of theorder O

(hL−1

), because L = (number of internal lines− number of vertices+ 1).

It is easy to extend these considerations to a loop with N vertices. Again, assumethat the pairs of external points which are joined to each vertex as well as theordering of vertices along the loop are specified; this corresponds to a well-definedsituation in momentum space. We have the factor (1/N !)(1/4!)N from the generalformula (2.89) and an extension of our discussion of the N = 2 case gives (3 ×4)N ×2N−1×N !. So we again get the symmetry factor 1

2 for a Feynman diagram inmomentum space, with momenta specified along all the internal lines, for example,as in Fig. 2.12. For N vertices there are (2N−1)(2N−3) . . . 1×(N−1)! differentgraphs in momentum space; the factor (2N − 1) . . . 1 gives the number of differentgroupings of 2N momenta in N pairs and (N − 1)! gives the number of theirdifferent orderings along the loop.

An important modification of the lagrangian (2.82) follows if we specify that allproducts of the field operators in it are normal-ordered

L = 12 :∂µ�(x)∂µ�(x): − 1

2 m2 :�2(x): − (λ/4!) :�4(x): (2.104)

This can be rewritten in a form similar to (2.82), but with a number of counterterms;remember that under the path integral the fields are c-numbers and the only wayof specifying their order is by introducing appropriate counterterms. From Wick’stheorem it follows that (see Section 2.7):

�2(x) = :�2(x): + const. (2.105)

�4(x) = :�4(x): + 6G(2)0 (x, x) :�2(x): + const. (2.106)

so that

:�4(x): = �4(x)− 6G(2)0 (x, x)�2(x)− const. (2.107)

and the other counterterms are constants. In the resulting perturbation expansion

100 2 Path integrals

Fig. 2.13.

the diagram is now cancelled by the−6G(2)0 (x, x)�2(x) counterterm.

Note, however, that the higher order corrections to this diagram, like that shownin Fig. 2.13, are not cancelled. The constant counterterms cancel some vacuumdiagrams.

2.5 Path integrals for fermions; Grassmann algebra

Anticommuting c-numbers

The path integral quantization method involves classical fields only and the theoryis formulated directly in terms of Green’s functions given by path integrals like(2.53). Since for Fermi fields we must have, for example,

〈0|T�(x1)�(x2)|0〉 = −〈0|T �(x2)�(x1)|0〉 (2.108)

the fermion classical fields in the path integral must be taken to be anticommutingquantities. The anticommuting c-number variables, or functions, are said to formGrassmann algebra.

There follows a brief introduction to the subject of Grassmann algebra. Let ustake η to be a Grassmann variable,

{η, η} = 0 or η2 = 0 (2.109)

Any function f (η) can be written as

f (η) = f0 + η f1 (2.110)

For instance if f (η) is taken to be an ordinary number, then f0 and f1 are ordinaryand Grassmann numbers, respectively. One defines the left and right derivatives bythe equation

d

dηη = η

←d

dη= 1 (2.111)

Therefore, when f1 is a Grassmann number,

d

dηf (η) = f1 = − f (η)

←d

dη(2.112)

2.5 Path integrals for fermions; Grassmann algebra 101

Furthermore, we define the integration by the requirement of linearity and thefollowing relations ∫

dη 1 = 0,∫

dη η = 1 (2.113)

The first relation follows from the requirement that the property of a convergentintegral over commuting numbers∫ ∞

−∞dx f (x) =

∫ ∞

−∞dx f (x + a)

valid for any finite a, holds for an integral over anticommuting numbers as well.The second relation is the normalization convention. We also add a specificationthat rule (2.113) applies when dη and η are next to each other. Comparing (2.113)with (2.111) we see that the integrals and the left derivatives are identical∫

dη f (η) = d

dηf (η) = f1 (2.114)

The integral of the derivative vanishes:∫dη

d

dηf (η) = d2

dη2f (η) = 0 (2.115)

Consider now the change of integration variable η → η′ = a + bη. One gets∫dη f (η) =

∫dη′

(dη

dη′

)−1

f (η(η′)) (2.116)

i.e. the standard Jacobian appears inverted. All the rules can be easily generalizedto the case of n real Grassmann variables. For a complex Grassmann variable thereal and imaginary parts can be replaced by η and η∗ as independent generators ofGrassmann algebra.

In the following chapter the Gaussian integral for complex Grassmann variableswill be shown to be very useful. It can be shown that instead of (2.76) we now have∫

dη1 dη∗1 . . . dηn dη∗n exp(η∗, Aη) = det A (2.117)

where

(η∗, Aη) =∑i, j

η∗i Ai jη j

Eq. (2.117) is valid for an arbitrary A. For a diagonal A the proof of (2.117) followsimmediately from the integration rules for Grassmann variables. The correspond-ing formula also holds for an integral over real variables (see Problem 2.4).


Gaussian integrals can be used to define the determinant of a matrix actingin superspace. Coordinates in superspace are commuting and anticommutingvariables, z and η, respectively. We can define a linear transformation in superspace(

z′

η′

)=[

A DC B

](zη

)≡ M

(zη

)where matrices A and B are commuting, and C and D are anticommuting. Vectorsin superspace are decomposed in terms of the variables z and η.

The superdeterminant of a superspace matrix M can be constructed by calculat-ing a generalized Gaussian integral

(det M)−1 =∫

dz dz∗

2πdη∗ dη exp[−(z∗, Az)− (z∗, Dη)− (η∗,Cz)− (η∗, Bη)]

To evaluate the integral we make a shift in integration variables

η = η′ − B−1Cz

η∗ = η∗ ′ − z∗DB−1

and using (2.76) and (2.117) we get

det M = det(

A − DB−1C)

det B

If we shift z and z∗ we get

det M = det A

det(B − C A−1 D

)These results for the superdeterminant are not surprising if we write

M =[

A 0C 1

][1 A−1 D0 B − C A−1 D

]or

M =[

1 D0 B

][A − DB−1C 0

B−1C 1

]Several other properties of the superdeterminant are listed in Problem 2.9.

Dirac propagator

The classical Dirac fields �(x) and �(x) are taken to be elements of aninfinite-dimensional Grassmann algebra. The generating functional W�

0 [α] for afree-fermion field described by the lagrangian

L� = �(i/∂ − m)�

2.5 Path integrals for fermions; Grassmann algebra 103

is

W�0 [α, α] = N

∫D� D� exp

{(i/h)

∫d4x [L�(x)+ hα(x)�(x)+ h�(x)α(x)]

}(2.118)

where

N−1 = 〈0|0〉 = W�0 [0, 0]

and α(x) and α(x) are sources of fermion fields. Using a formula analogous to(2.81) but for Grassmann fields∫

DηDη∗ exp[i(η∗, Aη)+ i(β∗, η)+ i(η∗, β)] = det(iA) exp[i(β∗,−A−1β

)](2.119)

we can rewrite W�0 [α, α] as follows:

W�0 [α, α] = exp

[ih∫

d4x d4 y α(x) S(x − y) α(y)

](2.120)

where

(i/∂ − m)S(x − y) = −1lδ(x − y) (2.121)

Therefore

S(x − y) = −∫

d4k

(2π)41l

/k + m

k2 − m2 + iεexp[−ik(x − y)] (2.122)

The Feynman propagator defined as

SF(x − y) = 〈0|T�(x)�(y)|0〉/〈0|0〉 =∫

D� D� �(x) �(y)

× exp

[(i/h)

∫d4x L�(x)

]/∫D� D� exp

[(i/h)

∫d4x L�(x)

](2.123)

reads

SF(x − y) =(

1

i

)2δ

δα(x)W�

0 [α, α]←δ

δα(y)

∣∣∣∣α=α=0

(2.124)

and using (2.120) one gets

SF(x − y) = −ihS(x − y) (2.125)

Imagine now a theory of interacting fermions with the action

S =∫

d4x L�(x)+∫

d4x �(x) V (x)�(x)

= S0 + SI (2.126)


Fig. 2.14.

Following Section 2.4 we can develop perturbation theory with the interaction termas the perturbation. In the expansion in powers of SI of, for example, the vacuum-to-vacuum amplitude (the denominator in (2.89)) we find under the path integralthe following products of the � and � fields:

∫dx1 . . . dxk �(x1) V (x1)�(x1) �(x2) V (x2)�(x2) . . . �(xk) V (xk)�(xk)

(2.127)which correspond to fermion loops like that shown in Fig. 2.14. The path integralwith insertion (2.127) can be calculated explicitly by differentiating W�

0 [α, α]given by (2.120) with respect to α and α. Before making the substitution

�(xi ) → 1

i

→δ

δα(xi ), �(xi ) → 1

i

←δ

δα(xi )(2.128)

we must, however, rearrange the fields as follows

V (x1)�(x1)�(x2)V (x2)�(x2)�(x3) . . . V (xk)�(xk)�(x1)

This reordering introduces the familiar minus sign for a closed fermion loop. From(2.120) and (2.128) we obtain for the closed fermion loop the following result

(−)V (x1)SF(x1 − x2)V (x2)SF(x2 − x3) . . . V (xk)SF(xk − x1) (2.129)

2.6 Generating functionals 105

2.6 Generating functionals for Green’s functions and proper vertices;effective potential

Classification of Green’s functions and generating functionals

There are three basic sorts of Green’s functions. The full Green’s functions aredefined by (2.54)

G(n)(x1, . . . , xn) = 〈0|T�(x1) . . . �(xn)|0〉/〈0|0〉=∫

D��(x1) . . . �(xn) exp{(i/h)S[�]}

×(∫

D� exp{(i/h)S[�]})−1

(2.54)

with appropriate generalization to the case of other fields; for simplicity, in thissection we shall always consider the case of a single field. In perturbation theorythe full Green’s functions G(n) are given by the sum of all diagrams with n externallegs, including disconnected ones, with the exception of the vacuum diagramswhich are cancelled by the normalization factor.

The connected Green’s functions G(n)conn(x1, . . . , xn) are defined as the connected

part of G(n). A connected Green’s function is obtained by disregarding all termswhich factorize into two or more functions with no overlapping arguments. For aFourier transform this implies that no subset of n external momenta is conservedseparately. The Feynman diagrams contributing to G(n)

conn are all connected.The connected proper vertex functions �(n)(x1, . . . , xn), also called the one-

particle-irreducible (1PI) Green’s functions, are given by the Feynman diagramswhich are one-particle-irreducible; i.e. they remain connected after an arbitraryinternal line is cut. In the lowest order (no loops) the connected proper vertexfunctions coincide when appropriately normalized with the vertices of the originallagrangian.

To each type of the Green’s functions we assign the corresponding generatingfunctional. These are defined as follows

W [J ] =∞∑

n=0

in

n!

∫d4x1 . . . d4xn G(n)(x1, . . . , xn) J (x1) . . . J (xn) (2.130)

Z [J ] = h∞∑

n=1

in−1

n!

∫d4x1 . . . d4xn G(n)

conn(x1, . . . , xn) J (x1) . . . J (xn) (2.131)

and

�[�] =∞∑

n=1

1

n!

∫d4x1 . . . d4xn �(n)(x1, . . . , xn)�(x1) . . . �(xn) (2.132)

Observe that in the generating functional �[�] we denote the arguments as �(x),


not J (x): this is convenient because of the relationship between �[�] and theaction integral S[�] which we shall discuss below. It follows from definitions(2.130)–(2.132) that

G(n)(x1, . . . , xn) =(

1

i

)nδnW [J ]

δ J (x1) . . . δ J (xn)

∣∣∣∣J≡0

(2.133)

G(n)conn(x1, . . . , xn) = 1

h

(1

i

)n−1δn Z [J ]

δ J (x1) . . . δ J (xn)

∣∣∣∣J≡0

(2.134)

and

�(n)(x1, . . . , xn) = δn�[�]

δ�(x1) . . . δ�(xn)

∣∣∣∣�≡0

(2.135)

The generating functional of the full Green’s functions has been studied inSections 2.4 and 2.5. It is given by the path integral formula

W [J ] =∫

D� exp

[i

h

{S[�] + h

∫d4x J (x)�(x)

}]/∫D� exp

{i

hS[�]

}(2.136)

We want to show that the functionals Z [J ] and �[�] are given by the followingrelations:

W [J ] = exp{(i/h)Z [J ]} (2.137)

and

�[�CL] = Z [J ] − h∫

d4x J (x)�CL(x) (2.138)

where the ‘classical’ field �CL(x) is defined as the field which minimizes thecombination

�[�] + h∫

d4x J (x)�(x)

or by the equationδ�[�]

δ�(x)

∣∣∣∣�=�CL

= −h J (x) (2.139)

By differentiating (2.138) with respect to J (x) we also find

δZ [J ]

δ J (x)=∫

d4 yδ�[�CL]

δ�CL(y)

δ�CL(y)

δ J (x)+ h�CL(x)+ h

∫d4 y

δ�CL(y)

δ J (x)J (y)

(2.140)

or, with (2.139),

�CL(x) = 1

h

δZ [J ]

δ J (x)= 〈0|�(x)|0〉J

〈0|0〉J(2.141)


Eq. (2.138) is the Legendre transform. If Z [J ] is known we can use (2.141) todetermine J (x) in terms of �CL(x) so that the r.h.s. of (2.138) can be written asa functional of �CL(x) which determines �[�CL]. The value of (2.141) when theexternal source is turned off (J (x) ≡ 0) is the vacuum expectation value of thefield �(x) which, for the time being, we assume to be zero:

δZ [J ]

δ J (x)

∣∣∣∣J≡0

= 0 (2.142)

The case of a non-zero field vacuum expectation value reflecting spontaneoussymmetry breaking will be discussed later in this section.

Eq. (2.139) expresses J (x) in terms of �CL(x) and in this sense it is the inverseof (2.141). In particular, together with (2.141) and (2.142), it implies that

δ�[�CL]

δ�CL(x)

∣∣∣∣�CL≡0

= 0 (2.143)

because when J ≡ 0, �CL takes the value 0 and vice versa.It remains to prove that (2.137) and (2.138) hold for the generating functionals

defined by (2.130)–(2.132). This can be checked by induction (see, for example,Abers & Lee (1973)). Here we shall use another argument which allows a simpleunderstanding of the relationship between �[�] and S[�].

Effective action

Consider the classical field equation (we take the case of the λ�4 theory fordefiniteness) (

∂µ∂µ + m2

)�c = −(λ/3!)�3

c + h J (x), λ > 0 (2.144)

Eq. (2.144) follows from the action principle

δ

δ�

∣∣∣∣�=�c

{S[�] + h

∫d4x J (x)�(x)

}= 0 (2.145)

(do not confuse �c and �CL), where

S[�] =∫

d4x[− 1

2�(∂µ∂

µ + m2)�− (λ/4!)�4

]Eq. (2.144) can be solved perturbatively. In the zeroth order in λ we obtain

�(0)c (x) =

∫d4 y [−ihG(x − y)]iJ (y) (2.146)

where G(x − y) is defined by (2.68)(∂µ∂

µ + m2)G(x − y) = δ(x − y) (2.68)


Fig. 2.15.

(the iε term should always be remembered). To obtain the first order correction weinsert �(0)

c in the r.h.s. of (2.144). The result reads

�(1)c (x) = �(0)

c (x)+∫

d4 y [−ihG(x − y)](−iλ/h3!)

{∫d4z [−iG(y − z)]iJ (z)

}3

(2.147)

which can be represented diagrammatically as shown in Fig. 2.15. Repeating theiterations we obtain the perturbation series for the classical field �c(x). It is easyto see that only tree diagrams appear and that they are all connected.

Consider now the generating functional W [J ], given by (2.136), in the classicalapproximation. This corresponds to replacing �(x) in the integrand of (2.136) bythe classical solution �c(x). Only the numerator then contributes because J ≡ 0in the denominator, and we get

W [J ] = exp

((i/h)

{S[�c] + h

∫d4x J (x)�c(x)

})(2.148)

Thus, if we assume (2.137), in the classical approximation Zc[J ] reads

Zc[J ] = S[�c] + h∫

d4x J (x)�c(x) (2.149)

and it is indeed given by the connected tree diagrams. This can be seen bycalculating δn Z/δ J (x1) . . . δ J (xn)|J≡0 using (2.149) and the expansion (2.147) forthe �c(x).

Moreover, from (2.138) and (2.141), in the classical approximation �[�CL] =S[�c] so in this approximation �[�c] as given by (2.138) is indeed the generatingfunction for the 1PI Green’s functions (because δn S[�c]/δ�c(x1) . . . δ�c(xn)|�c=0

generates proper vertex functions in the tree approximation).When the loop corrections are taken into account the connected Green’s func-

tions as well as the generating functional Z [J ] can still be understood as givenby the tree diagrams but with the elementary vertices replaced by 1PI Green’sfunctions. If in (2.136) we replace the action S[�] by the functional �[�] andinstead of doing the path integration exactly, again make in the integrand the‘classical’ approximation �(x) → �CL(x) with �CL(x) defined by (2.139), we


shall obtain the exact result. This is because all loop corrections are alreadyincluded in �[�]. In this sense �[�] can be treated as the effective actionincluding the loop corrections. �(n)(x1, . . . , xn)s are n-point non-local verticesof this effective action. We conclude that to all orders

W [J ] = exp

[(i/h)

{�[�CL] + h

∫d4x J (x)�CL(x)

}](2.150)

and therefore, as �[�CL] and �CL are both given by connected diagrams, Z [J ]given by (2.137) is indeed connected (the connected diagrams exponentiate). Also(2.138) holds to all orders.

It is worth observing explicitly that differentiating (2.141) with respect to�CL(y), using (2.139) and setting �CL(x) = 0 one gets∫

d4z �(2)(x − z) G(2)conn(z − x) = ihδ(x − y) (2.151)

or, in momentum space,

�(2)(p)G(2)(p) = ih (2.152)

(we draw attention to the factor i). For higher n, strictly speaking, i�(n)s are the1PI functions given directly in terms of Feynman diagrams.

Spontaneous symmetry breaking and effective action

The formalism of the effective action can be extended to the case of a non-vanishing field vacuum expectation value which corresponds to spontaneoussymmetry breaking (the latter is discussed in detail in Chapter 9). The functionalsW [J ] and Z [J ] are taken as given by (2.136) and (2.137), respectively. However,we now suppose that (2.142) is replaced by

〈0|�(x)|0〉〈0|0〉 = 1

h

δZ [J ]

δ J (x)

∣∣∣∣J≡0

= v (2.153)

where v = const. from translational invariance. It can be proved, for example, byinduction, that higher derivatives of Z [J ] at J ≡ 0 are connected Green’s functionsof the field � = �− v whose vacuum expectation value vanishes:

1

h

(1

i

)n−1δn Z [J ]

δ J (x1) . . . δ J (xn)

∣∣∣∣J≡0

= 〈0|T �(x1) . . . �(xn)|0〉 (2.154)

The effective action is defined by (2.138) and (2.139), so that

δ�[�CL]

δ�CL(x)

∣∣∣∣�CL=v

= 0 (2.155)


This follows from (2.139), (2.141) and (2.153) because when J ≡ 0, �CL takes thevalue v and vice versa. Also, we can prove, for example, by induction, that �[�CL]given by (2.138) generates 1PI vertices for the field � = �− v

δn�[�CL]

δ�CL(x1) . . . δ�CL(xn)

∣∣∣∣�CL=v

= �(n)�=�−v

(x1, . . . , xn) (2.156)

and

�[�CL] =∞∑

n=1

1

n!

∫d4x1 . . . d4xn �

(n)�=�−v

(x1, . . . , xn)

×(�CL(x1)− v) . . . (�CL(xn)− v) (2.157)

It is important to observe that even for 〈0|�(x)|0〉 = v �= 0 the effective action�[�CL] can also be expanded in terms of the original fields �(x), see (2.132). Thecoefficients of such an expansion

δn�[�CL]

δ�CL(x1) . . . δ�CL(xn)

∣∣∣∣�CL≡0

= �(n)� (x1, . . . , xn) (2.158)

are the 1PI Green’s functions for the original fields �(x). This can be proved, forexample, by repeating the inductive arguments leading to (2.156) but for �CL = 0and correspondingly for J (x) = J0, where

δZ [J ]

δ J (x)

∣∣∣∣J≡J0

= 0 (2.159)

One can first check by induction that higher derivatives of Z [J ] taken for J ≡ J0

give connected Green’s functions 〈0|T� . . .�|0〉conn. Then we differentiate theappropriate number of times the relation δZ [J ]/δ J (x) = �CL(x) with respect to�CL and express the derivatives δn J (x)/δ�CL(x1) . . . �CL(xn) in terms of �[�CL]defined by (2.138) and (2.139). In the limit J ≡ J0 and �CL ≡ 0 we get the desiredresult. The relation between the Green’s functions �

(n)�=�−v

and �(n)� is given by the

analogue of the relation(d

dx

)n

f (x)|x=a =∞∑

m=0

am

m!

(d

dx

)n+m

f (x)|x=0

Thus we have

�(n)�=�−v

(x1, . . . , xn) =∞∑

m=0

(1/m!)vm∫

d4xn+1 . . . d4xn+m

×�(n+m)� (x1, . . . , xn, . . . , xn+m) (2.160)


or, in momentum space,

�(n)�

(p1, . . . , pn) =∞∑

m=0

(1/m!)vm�(n+m)� (p1, . . . , pn, 0, . . . , 0︸︷︷︸

m

) (2.161)

where �(n)(p1, . . . , pn) is the Fourier transform of the proper vertex�(n)(x1, . . . , xn)

�(n)(p1, . . . , pn)(2π)4δ(p1 + · · · + pn) =∫ n∏

i=1

d4x exp(ipi xi ) �(n)(x1, . . . , xn)

Eq. (2.161) expresses the 1PI Green’s functions in a theory with spontaneouslybroken symmetry in terms of the 1PI Green’s functions calculated as in thesymmetric mode.

Effective potential

Eq. (2.155) has very important implications in studies of spontaneous symmetrybreaking: it tells us that the field vacuum expectation value is the value of �CL(x)which extremizes �[�CL]. These implications can be discussed more convenientlyif we introduce the so-called effective potential V (ϕ). It is defined by the equation

�[�(x) = ϕ = const.] = −(2π)4δ(0)V (ϕ) (2.162)

(V (ϕ) is a function, not a functional). Condition (2.155) now translates into

dV

dϕ

∣∣∣∣ϕ=v

= 0 (2.163)

which is more useful than the original one because the effective potential V (ϕ) canbe calculated perturbatively in a systematic way. Using representation (2.132) forthe �[�] we get

V (ϕ) = −∞∑

n=2

(1/n!)�(n)� (0, 0, . . . , 0)ϕ . . . ϕ (2.164)

so that the nth derivative of V at ϕ = 0 is the sum of all 1PI diagrams forthe original fields of the lagrangian, with n vanishing external momenta. (Inthe tree approximation V is just the ordinary potential, the negative sum of allnon-derivative terms in the lagrangian density.) Alternatively, one can rewrite thelagrangian of the theory in terms of shifted fields � = � − v considering v asan extra free parameter, develop the appropriate Feynman rules and fix v directlyfrom (2.155) and (2.156), i.e. by demanding that all tadpole diagrams sum to zero.This method of calculation is equivalent to computing directly the derivative of Vand demanding that it vanishes, without computing V first, but it makes use of the


Fig. 2.16.

formulation of the theory in which the underlying spontaneously broken symmetryis not explicit (see Problem 4.1).

Knowledge of the effective potential means that one knows the structure ofspontaneous symmetry breaking. In principle, using (2.164), we can calculate theeffective potential in perturbation theory but the calculation involves a double sum:over all 1PI Green’s functions and for each 1PI Green’s function there is the expan-sion in powers of the coupling constant. A sensible way of organizing this doubleseries is the loop expansion. It has been seen at the end of Section 2.4 that loopexpansion is an expansion in powers of Planck’s constant h: a Feynman diagramwith L loops is O

(hL−1

). Thus, the loop expansion preserves any symmetry of

the lagrangian and it is unaffected by shifts of fields and by the redefinition of thedivision of the lagrangian into free and interacting parts associated with such shifts.Indeed expansion in powers of h is an expansion in a parameter that multiplies thetotal Lagrange density; (1/h)L→ L corresponds to a change of units into h = 1.

The calculation of the effective potential for the λ�4 theory in the one-loopapproximation is simple. To the lowest order (tree approximation) the only non-vanishing 1PI Green’s functions are �(2)(p, p) = p2 −m2 and �(4) = −λ, and weget

V0(ϕ) = + 12 m2ϕ2 + (λ/4!)ϕ4 (2.165)

To the next order (one-loop approximation) we have the infinite series of diagramsshown in Fig. 2.16 which gives the following contribution

V1(ϕ) = i∞∑

n=1

1

2n

∫d4k

(2π)4

[λ

1

k2 − m2 + iε

ϕ2

2

]n

= − 12 i∫

d4k

(2π)4ln

(1 −

12λϕ

2

k2 − m2 + iε

)(2.166)

The factor (1/2n)(

12

)ncan be easily understood by recalling the discussion at the

end of Section 2.4: each diagram represents [(2n − 1)(2n − 3) . . . 1](n − 1)!Feynman diagrams each with the symmetry factor 1

2 ; thus we get (1/2n!)[(2n −1)(2n − 3) . . . 1](n − 1)! 1

2 = (1/2n)(

12

)n. The integral in (2.166) exhibits the

UV divergence typical for perturbative calculations in quantum field theories.

2.7 Green’s functions and the scattering operator 113

Its evaluation requires setting up the renormalization programme and for furtherdiscussion of V1 given by (2.166) we refer the reader to Section 11.2.

Our final, but very important, remark in this section deals with the physicalmeaning of the effective potential. It can be shown (see, for example, Coleman(1974)) that V (ϕ) is the expectation value of the energy density in a certain statefor which the expectation value of the field is ϕ. This interpretation of V (ϕ) isnot surprising: it is already suggested by the classical limit of V (ϕ) and by, forexample, (2.163) for the field vacuum expectation value 〈0|�(x)|0〉 = v. If V (ϕ)

has several local minima, it is only the absolute minimum that corresponds to thetrue ground state of the theory.

2.7 Green’s functions and the scattering operator

The Green’s functions of quantum field theory can be used to determine the matrixelements of the operator S which are directly related to scattering amplitudesand therefore experimental results. Formally, the operator S can be obtained asan infinite time limit of the evolution operator in the interaction picture. In theinteraction picture the time evolution of operators is governed by the free, H0, partof the complete hamiltonian operator H = H0 + HI of the system (from now onwe put h = 1)

�(x, t) = exp(iH0t)�(x, 0) exp(−iH0t) (2.167)

while the state vectors obey the equation

i∂

∂t

∣∣∣∣t⟩ = HI(t)|t〉; hI(t) = exp(iH0t)HI exp(−iH0t) (2.168)

with the formal solution

|t〉 = U (t, t ′)|t ′〉 (2.169)

U (t, t ′) = T exp

[−i∫ t

t ′dt HI(t)

](2.170)

We then define the scattering operator to be

S = U (∞,−∞) = T exp

[−i∫ ∞

−∞dt HI(t)

](2.171)

It transforms the ‘incoming’ (t → −∞) states into the ‘outgoing’ (t → +∞)ones.

The interaction hamiltonian HI(t) in (2.170) can be expressed in terms of theinteraction picture field operators. For simplicity, let us consider the case of asingle scalar field, described by the lagrangian density

L = 12

(∂µ�∂µ�− m2�2

)− V (�) (2.172)


or, equivalently, by the hamiltonian

H = 1

2

∫d3x

{�2(x, t)+ [∇∇∇�(x)]2 + m2�2(x)

}+ ∫d3x V (�(x)) = H0 + HI

(2.173)where

�(x, t) = ∂L/∂�(x, t).

Eq. (2.167) implies that the interaction picture field operator obeys the Klein–Gordon equation of a free field(

�+ m2)�(x) = 0 (2.174)

so that at any moment of time we can write

�(x, t) =∫

d3k

(2π)32k0[a(k) fk(x, t)+ a†(k) f ∗k (x, t)] (2.175)

where fk are the plane wave solutions of the Klein–Gordon equation

fk(x, t) = exp[−i(k0t0 − k · x)], k0 = +(k2 + m2)1/2

(2.176)

and a(k), a†(k) are the annihilation and creation operators of scalar particleswith three-momentum k satisfying [a(k), a†(k′)] = (2π)32k0δ(k − k′) (seeSection 1.4). We can now use (2.171), (2.173) and (2.175) to express S in terms ofthe annihilation and creation operators and calculate the matrix elements betweenstates containing scalar particles.

When calculating the matrix elements it is convenient to deal with normal-ordered products (in the interaction picture), with all creation operators standingto the left of the annihilation operators.

This is done by using Wick’s theorem. When applied to the T -ordered productof m field operators Wick’s theorem has the well-known form (Bjorken & Drell1965)

T�(x1) . . . �(xm) = :�(x1) . . . �(xm): +∑i, j

i �= j

(−i)G(xi − x j ) :∏

k �=i, j

�(xk): + · · ·

+∑i1, j1

. . .∑in , jn

(−i)G(xi1 − x j1) . . . (−i)G(xin − x jn )

×{

1 when m = 2n

:�(xl): when m = 2n + 1, l �= i1 �= · · · �= jn(2.177)


A more general and compact formulation is

T F[�] = exp

[− 1

2 i∫

δ

δ�(x)G(x − y)

δ

δ�(y)d4x d4 y

]:F[�]: (2.178)

where F[�] is some functional of the field operator � and G(x − y) is the Green’sfunction given by (2.91). Functional differentiation with respect to the operator� has been introduced here. It is a straightforward generalization of the usualfunctional differentiation: the only difference is that one must keep in mind thatthe field operators for different values of time do not commute. If we assumeF[�] to be a product of four field operators, �(x1)�(x2)�(x3)�(x4), and take thematrix element of (2.178) between the vacuum states, the result is exactly (2.94),the corollary of Wick’s theorem used to derive the perturbation theory Feynmanrules in the previous section.

Now let us take the interaction hamiltonian to be

HI(t) =∫

d3x V (�) = −∫

d3x J (x)�(x) (2.179)

where J (x) is a c-number function. This corresponds to the case of scatteringfrom an external source. We obtain, after performing the functional differentiationin (2.178):

T exp

(i∫

d4x J�

)= exp

[12 i∫

d4 y d4x J (x) G(x − y) J (y)

]× :exp

(i∫

d4x J�

):

= W0[J ] :exp

(i∫

d4x J�

): (2.180)

where W0[J ] is the previously introduced generating functional of the free-fieldGreen’s functions. We observe that

J (x)W0[J ] = −i∫

d4 y G−1(x − y)δ

δ J (y)W0[J ] (2.181)

so that (2.180) can be rewritten as follows

T exp

(i∫

d4x J�

)= : exp

[ ∫d4x d4 y �(x) G−1(x − y)

δ

δ J (y)

]: W0[J ]

(2.182)Here G−1(x − y) is the inverse operator to G(x − y)∫

d4 y G−1(x − y) G(y − z) = δ(x − z) (2.183)

i.e. the Klein–Gordon operator.


The generalization to arbitrary V (�) follows from the formula

T exp

[−i∫

d4x V (�)

]= exp

[−i∫

d4x V

(1

i

δ

δ J

)]T exp

(i∫

d4x J�

)∣∣∣∣J≡0

(2.184)

Consequently, the S operator of the interacting scalar field can be written in theform

S = T exp

[−i∫

d4x V (�)

]= : exp

[ ∫d4x d4 y �(x) G−1(x − y)

δ

δ J (y)

]:

× exp

[−i∫

d4x V

(1

i

δ

δ J

)]W0[J ]

∣∣∣∣J≡0

= N−1 :exp

[ ∫d4x d4 y �(x) G−1(x − y)

δ

δ J (y)

]: W [J ]

∣∣∣∣J≡0

(2.185)

We have been dealing with fields in the interaction picture but in W [J ] we canrecognize, by (2.88), the generating functional of the previously introduced Green’sfunctions of the interacting theory, defined as the vacuum matrix elements of theT -ordered products of the field operator in the Heisenberg picture; see (2.54).Expanding the exponential we obtain

S = N−1∞∑

n=0

1

n!

∫d4x1 d4 y1 . . . d4xn d4 yn G(n)(y1, . . . , yn)

× iG−1(x1 − y1) . . . iG−1(xn − yn) :�(x1) . . . �(xn): (2.186)

The factor N−1 contains all the vacuum-to-vacuum transitions and will be omittedin the following.

The calculation of matrix elements is now straightforward. Let the initialstate involve n particles and the final state m particles. We have (c, c′ are thenormalization constants)

|k1 . . .kn〉 = ca+(k1) . . . a+(kn)|0〉〈k′1 . . .k′m | = c′〈0|a(k′1) . . . a(k′m)

}(2.187)

It is convenient to assume that k′i �= k j for all i, j pairs. Only the term in the serieswith :�(x1) . . . �(xn+m): will then contribute. Of the (n+m) field operators n willact as the annihilation and m as the creation operators: this gives a combinatorial

factor

(n + m

m

). Finally, there are n!m! ways of matching the annihilation and


creation operators with the external particles. The result is

〈k′1 . . .k′m |S|k1 . . .kn〉 = cc′∫

d4x1 . . . d4xn+m f ∗ ′k1(x1) . . . fkn (xn+m)

×∫

d4 y1 . . . d4 yn+m iG−1(x1 − y1) · · ·×iG−1(xn+m − yn+m)G

(n+m)(y1, . . . , yn+m) (2.188)

With the usual invariant normalization

|k1 . . .kn〉 = a†(k1) . . . a†(kn)|0〉 (2.189)

this is just the on-shell limit of the Fourier transformed function

〈k′1 . . .k′m |S|k1 . . .kn〉 = limk2

i →m2

∏i

[1

i

(k2

i − m2)]

×G(n+m)(k ′1, . . . , k ′m,−k1, . . . ,−kn) (2.190)

where

G(n)(k1, . . . , kn) =∫

d4 y1 exp(ik1 y1) . . .

∫d4 yn exp(ikn yn) G(n)(y1 . . . yn)

(2.191)We see that the S matrix element is equal to the multiple on-shell residue of theFourier transformed Green’s function. Its connected part is proportional to theon-shell limit of the so-called connected truncated Green’s function, obtained fromG(n)

conn by factorizing out the exact two-point functions G(2)(ki ) for each of theexternal lines of G(n)

conn.At this point we must observe that in the derivation of (2.190) we have made the

assumption (following from (2.175)) that the one-particle state created by �(x)from the vacuum has the standard normalization

〈k|�(x)|0〉 = exp[i(k0x0 − k · x)] (2.192)

with 〈k| normalized as in (2.189), i.e.

〈k| = 〈0|a(k) (2.193)

or

〈k|k′〉 = (2π)32k0δ(k − k′)

Normalization (2.192) holds for the field operators both in the interaction and inthe Heisenberg pictures. Fields satisfying (2.192) we shall call the ‘physical’ fields.More general cases will be discussed in Chapter 4.


The above formal considerations leading to the Lehman–Symanzik–Zimmermann reduction theorem (2.190) can be supplemented by the followingarguments, which by themselves can actually be taken as a proof of (2.190). Let usconsider the Fourier transform (2.191) of the Green’s function G(n)(y1, . . . , yn) =〈0|T�(y1) . . . �(yn)|0〉, where we have taken 〈0|0〉 = 1, and assume that thefield �(x) is normalized according to (2.192). Among the various terms whichcontribute to G(n)(k1, . . . , kn) there is a time-ordering in which �(y1) standsfurthest to the left. We can insert a complete set of intermediate states between�(y1) and the remaining time-ordered product getting

G(n)(k1, . . . , kn) =∫

m

∑d4 y1 . . . d4 yn exp(ik1 y1) . . . exp(ikn yn)

×〈0|�(y1)|m〉〈m|T�(y2) . . . �(yn)|0〉�(y0

1 − max{

y02 . . . y0

n

})+ · · ·(2.194)

The y1-integration can now be carried out using

〈0|�(y1)|n〉 = 〈0|�(0)|n〉 exp(−ipn y1)

and the integral representation for the �-function

�(y0

1 − t) = i

2π

∫ ∞

−∞

dω exp[−iω

(y0

1 − t)]

ω + iε

One then gets the following∫d4 y1 exp[i(k1 − pn)y1]�

(y0

1 − t)

= iδ(k1 − pn)(2π)3 1

k01 − p0

n + iεexp

[i(k0

1 − p0n

)t]

= i(2π)3δ(k1 − pn)k0

1 + p0n

k21 − p2

n + iεexp

[i(k0

1 − p0n

)t]

(2.195)

Thus, a single particle state of mass m contributes a pole at k21 = m2 while no

other state |n〉 and none of the remaining terms arising from other time-orderingscan have such a pole. It is important to observe that the pole arises due to the limity0

1 →+∞ in the integral (2.195). We conclude therefore that

limk2

1→m2G(n)(k1, . . . , kn) = i

k21 − m2

〈k1, t = +∞|T�(y2) . . . �(yn)|0〉 (2.196)(∫|1〉〈1| ≡

∫d3 p

(2π)32p0|p〉〈p|

)


Similarly,

limk2

1→m2G(n)(−k1, . . . , kn) = i

k21 − m2

〈0|T�(y2) . . . �(yn)|k1, t = −∞〉 (2.197)

By repeating these steps we prove (2.190), remembering that

〈k′1 . . .k′m |S|k1 . . .kn〉 = 〈k′1 . . .k′m, t = +∞|k1 . . .kn, t = −∞〉As an explicit example illustrating (2.190) consider elastic scattering of two

identical scalar particles in the �4 theory. In the lowest order, using (2.102) weget

(S − 1)2→2 = −iλ(2π)4δ(4)(p1 + p2 − k1 − k2) (2.198)

The basic result, (2.188) and (2.190), is valid for both the path integral andcanonical formulations of quantum field theory. Perturbation theory for calculatingthe Green’s functions present in (2.188) and (2.190) can also be developed eitherin the path integral approach, as in Section 2.4, or in the (canonical) operatorformalism (see, for instance, Bjorken & Drell (1965), Weinberg (1995)). Thelatter approach is sometimes quite convenient, in particular for quick tree levelmanipulations. The S-matrix elements can be directly calculated as matrixelements of the operator in (2.171) (expressed in terms of fields satisfying freeequations of motion) taken between free particle states. For instance, for 2 → 2scattering in the �4 theory we have:

S2→2 = 〈p1, p2|S|k1, k2〉 (2.199)

= 〈0|a(p1)a(p2)T exp

{−i

λ

4!

∫d4x :�4:

}a†(k1)a

†(k2)|0〉

In the lowest order the contribution to the transition matrix is

〈p1, p2|S − 1|k1, k2〉 = −iλ

4!

∫d4x 〈0|a(p1)a(p2) :�4(x):a†(k1), a†(k2)|0〉

= −iλ

4

∫d4x 〈0|a(p1)a(p2)�

+(x)�+(x)�−(x)�−(x)

×a†(k1)a†(k2)|0〉 (2.200)

where �+ (�−) stands for positive (negative) frequency parts of the field operators(1.126). In this order the terms in (2.177) that contain contractions do notcontribute to the connected amputated part of the matrix element. Commutingthe creation and annihilation operators (using (1.129), (1.130)) and performing theintegral over d4x as well as the integrals over momenta in the field operators (1.126)we rederive (2.198).


As a second example consider the so-called four-fermion interaction lagrangianof the generic type

LI = −g :(� f4(x)�A� f3(x))(� f2(x)�B� f1(x)): (2.201)

(where �A(B) are some constant matrices in the spinor space) and compute thematrix elements of S−1 for the process f1 f2 → f3 f4. In the lowest order we have

〈 f4(p4, s4), f3(p3, s3)|S − 1| f2(p2, s2), f1(p1, s1)〉= −ig

∫d4x 〈0|b4d3 :

(�

†f4(x)γ 0�A� f3(x)

)(�

†f2(x)γ 0�B� f1(x)

): d†

2 , b†1|0〉

(2.202)

where we have used the abbreviations b4 ≡ b(p4, s4) etc. Using the Wick theorem(2.177) and repeating the same steps as in the previous example one arrives at

(S − 1) f1 f2→ f3 f4= −ig

∑s′i

∫d3q4

(2π)32Eq4

. . .

∫d3q1

(2π)32Eq1

×∫

d4x exp[−ix · (q1 + q2 − q3 − q4)]

×〈0|b4d3 :[u(q4, s ′4)b†(q4, s ′4)�Bv(q3, s ′3)d

†(q3, s ′3)]

×[v(q2, s ′2)d(q2, s ′2)�Au(q1, s ′1)b(q1, s ′1)]: d†2 , b†

1|0〉= −ig(2π)4δ(p4 + p3 − p2 − p1)

×[u(p4, s4)�Bv(p3, s3)][v(p2, s2)�Au(p1, s1)] (2.203)

It follows from this example that the initial (final) Dirac fermion with momentumk and spin s is described by the wave-function u(k, s) (u(k, s)) and the initial(final) antifermion with momentum k and spin s by v(k, s) (v(k, s)). The operatormethod is convenient to reach this conclusion quickly. Also, it is clear that theabsolute sign of the amplitude containing more than one fermion (or antifermion)in the initial and/or final states is purely conventional as it depends on the arbitraryordering of fermionic creation operators defining these states. The relative signs ofseveral different contributions to the transition amplitudes (if present) and variouscombinatorial factors are easy to fix by operator manipulations.

Problems

2.1 Derive (2.12) from (2.7), integrating over pi s first.

2.2 For a scalar field theory defined by a lagrangian

L = 12∂

µ�(x)∂µ�(x)− V (�(x))

Problems 121

and the corresponding equation of motion

��+ V ′(�) = 0

derive the equation of motion for the Green’s function 〈0|T�(x)�(y)|0〉:�x 〈0|T�(x)�(y)|0〉 = −〈0|T V ′(�(x))�(y)|0〉 − iδ(4)(x − y)

using definitions (2.54) and (2.56) and the invariance of the generating functional(2.57) under an infinitesimal change of variables

�(x) → �(x)+ ε f (x)

where f (x) is a function of x∫D� exp

{iS[�+ ε f ]+ i

∫d4x (�+ ε f ) J

}=∫

D� exp

{iS[�]+ i

∫d4x �J

}Repeat the same in Euclidean space.

2.3 Using the rules of Section 2.4 calculate the following symmetry factors S in the λ�4

theory

2.4 For real Grassmann variables ηi prove the formula∫dη1 . . . dηn exp

[ 12 (η Aη)

] = (det A)1/2

for any antisymmetric matrix A, where (η, Aη) =∑i, j ηi Ai jη j .

2.5 Prove (2.154), (2.156) and (2.158).

2.6 Prove (2.178).

2.7 Using representation (2.10) calculate the Green’s function for the harmonic oscillator

〈x ′| exp[−i(t ′ − t)H ]|x〉 =(

ω

2iπ sinω(t ′ − t)

)1/2

exp[iS(t ′ − t)]

where

H = − 12

(∂2/∂x2)+ 1

2ω2x2

S(τ ) = ω

2 sinωτ

[(x ′ 2 + x2) cosωτ − 2x ′x

]


Note that S(τ ) is the classical action (along the classical path).

2.8 For a scalar field theory as in Problem 2.2 with V (�) = λ�4 calculate the effectivepotential �[�] by evaluating the generating functional WE[J ] in Euclidean space

WE[J ] = NE

∫D� exp

{−SE[�] +

∫J� d4 x

}by the method of steepest descent. The steps are:

(a) Expand the exponent around �0 (often called the background field) which is thesolution of the classical equation of motion in the presence of the external sourceJ (x) (−∂µ∂

µ + m2)�0 + V ′(�0)− J = 0 (2.204)

to write

WE[J ] = NE exp

{−SE[�0] +

∫J�0 d4 x

}×∫

D� exp

{−1

2

∫ ∫d4 x d4 y

δ2SE[�0]

δ�(x)δ�(y)

× [�(x)−�0(x)][�(y)−�0(y)] + · · ·}

(2.205)

(b) In the classical approximation

WE[J ] = NE exp

{−SE[�0] +

∫d4 x J�0

}Evaluate Z [J ] and �[�] explicitly by solving (2.204) perturbatively in powers ofλ. Check that in this approximation �[�] is the classical action.

(c) Calculate �[�] in the saddle point approximation. Evaluate the Gaussian integralin (2.205) by using det A = exp(Tr ln A) and expanding the logarithm in powersof λ. Calculate the effective potential and prove (2.166).

2.9 For a matrix M acting in superspace(zη

)′= M

(zη

)=(

A DC B

)(zη

)where z and η are commuting and anticommuting coordinates, respectively:

(a) Write det M instead of (det M)−1 as a Gaussian integral.(b) Show that the supertrace defined as

Tr M = Tr A − Tr B

satisfies the cyclicity property

Tr[M1 M2] = Tr[M2 M1]

Problems 123

(c) Show that

Tr ln M1 M2 = Tr ln M1 + Tr ln M2

Hint: use the Campbell–Backer–Hausdorff formula for ordinary matrices

exp(A) exp(B) = exp

{A + B + 1

2![A, B] + 1

3!

( 12 [[A, B], B] + 1

2 [A, [A, B]])

+ 1

4![[A, [A, B]], B] + · · ·

}The exponent and the logarithm of a matrix in superspace are defined by a seriesexpansion, as for ordinary matrices.

(d) Writing

M =[

A 0C 1

][1 A−1 D0 B − C A−1 D

]and using property (c), prove that

ln det M = Tr ln M

and

det(M1 M2) = det M1 det M2

(e) Write the Jacobian of a transformation in superspace.

3

Feynman rules for Yang–Mills theories

3.1 The Faddeev–Popov determinant

Gauge invariance and the path integral

As discussed in the Introduction gauge quantum field theories are of specialphysical interest in our attempts to describe fundamental interactions. In thischapter we shall derive Feynman rules for gauge theories. Consider a path integralover gauge field Aµ, corresponding to a physical, i.e. gauge-invariant, quantityf (Aµ) ∫

DAµ f (Aµ) exp

(i∫

d4x L)

(3.1)

For brevity we write Aµ instead of Aαµ, where α is the gauge group index; L is the

lagrangian density. The integration measure DAµ we assume to be invariant undergauge transformations: i.e. it must have the following property:

DAµ = DAgµ

where g is an arbitrary transformation from the gauge group. Agµ denotes the

result of this transformation when applied to Aµ. The simplest formal ansatz forthe integration measure which has this property is DAµ = ∏

µ,α,x dAαµ(x), the

straightforward generalization of the integration measure introduced previously forscalar fields.

As explained before, the path integral in (3.1) runs over all possible config-urations Aµ(x), which implies multiple counting of the physically equivalentconfigurations (equivalent up to a gauge transformation). Let us divide theconfiguration space {Aµ(x)} into the equivalence classes {Ag

µ(x)} called the orbitsof the gauge group. An orbit of the group includes all the field configurations whichresult when all possible transformations g from the gauge group G are applied toa given initial field configuration Aµ(x). This construction can be representedgraphically as shown in Fig. 3.1. The integrand of (3.1) is constant along any orbit

124

3.1 The Faddeev–Popov determinant 125

Fig. 3.1.

of the gauge group. Consequently, the integral as it stands is proportional to aninfinite constant (the volume of the total gauge group G). This is not an importantdifficulty in itself, because the infinity can always be cancelled by a normalizationconstant. The problem appears when we want to calculate (3.1) perturbativelybecause local gauge symmetry implies that the quadratic part of the gauge fieldaction density has zero modes and therefore cannot be inverted in the configurationspace {Aµ(x)} so that the propagator of the gauge field cannot be defined. One wayto resolve this problem is to apply perturbation theory to the functional integralover the coset space of the orbits of the group (in other words, over the physicallydistinct field configurations) which follows from (3.1) after the infinite constanthas been factorized out. We shall now describe a procedure by means of which thisfactorization can be achieved.

Let Dg denote an invariant measure on the gauge group G

Dg = D(gg′); Dg =∏

x

dg(x) (3.2)

and let us introduce a functional [Aµ] defined by the following equation

1 = [Aµ]∫

Dg δ[F[Ag

µ

]](3.3)

Here δ[ f (x)] represents the product of the usual Dirac δ-functions:∏

x δ( f (x)),one at each space-time point. As for the functional F[Aµ], we assume that theequation

F[Ag

µ

] = 0 (3.4)

has exactly one solution, g0, for any initial field Aµ. In the configuration space{Aµ(x)} the equation F[Aµ] = 0 must then define a surface that crosses any of theorbits of the group exactly once, i.e. (3.4) defines a ‘gauge’. Let us observe that thefunctional F[Aµ] should be of the general form Fα(x, [Aµ]), where α is the groupindex of the adjoint representation and x a space-time point. In fact to fix a gaugewe need (at each space-time point) one equation for each parameter of the group.

126 3 Feynman rules for Yang–Mills theories

[Aµ] is invariant under gauge transformations. This can be shown as follows

−1[Ag

µ

] = ∫Dg′ δ

[F[Agg′

µ

]] = ∫D(gg′) δ

[F[Agg′

µ

]]=∫

Dg′′ δ[F[Ag′′

µ

]] = −1[Aµ] (3.5)

where we have used the invariance property of the group measure Dg. In effect,the functional [Aµ] depends only on the orbit to which Aµ belongs.

Our aim is to replace integration over all field configurations by integrationrestricted to the hypersurface F[Aµ] = 0. In this case each orbit will contributeonly one field configuration and we shall have an integration over physicallydistinct fields. We start by inserting (3.3) (which is equal to one) under the pathintegral in (3.1). Next we change the order of integrations. The result is∫

Dg∫

DAµ [Aµ] f (Aµ) δ[F[Ag

µ

]]exp{iS[Aµ]} (3.6)

An important observation is that the complete expression under the∫Dg integral is

in fact independent of g. To show this we can use the gauge invariance of∫DAµ,

[Aµ], f (Aµ) and S[Aµ] to replace them by∫DAg

µ, [Ag

µ

], f(

Agµ

)and S

[Ag

µ

]:

the result ∫DAg

µ [Ag

µ

]f(

Agµ

)δ[F[Ag

µ

]]exp

{iS[Ag

µ

]}can be made manifestly g-independent by a change in notation: Ag

µ → Aµ.Consequently, the group integration

∫Dg factorizes out to produce an infinite

constant: the volume of the full gauge group. We obtain finally(∫Dg

)∫DAµ [Aµ] δ[F[Aµ]] f (Aµ) exp{iS[Aµ]} (3.7)

which defines the theory as formulated in the F[Aµ] = 0 gauge. Unlike (3.1), (3.7)can be used as a starting point for perturbative calculations.

Faddeev–Popov determinant

We have still to calculate [Aµ]. From (3.3) we obtain, after formal manipulations

−1[Aµ] =∫

DF

(det

δF[Ag

µ

]δg

)−1

δ[F]

i.e.

[Aµ] = detδF[Ag

µ

]δg

∣∣∣∣F[Ag

µ]=0

(3.8)


[Aµ] is usually called the Faddeev–Popov determinant. It is convenient to use thegauge invariance property of [Aµ] to choose Aµ which already satisfies the gaugecondition F[Aµ] = 0. Then in (3.8) we can replace the constraint F

[Ag

µ

] = 0 byg = 1, which simplifies the practical calculations

[Aµ] = detδF[Ag

µ

]δg

∣∣∣∣g=1

; F[Aµ] = 0 (3.9)

Near g = 1 we only have to deal with the infinitesimal transformations: U (�) =1 − iT α�α(x) (where �α(x) � 1) and the invariant group measure Dg takes thesimple form D� ≡ ∏

α,x d�α(x). It is easy to check the invariance property ofD�

D� = D�′′ (3.10)

where U (�′′) = U (�)U (�′) and �′ is arbitrary. For infinitesimal transformations�′′ ≈ �′ +� and therefore det(d�′′/d�) = 1, so that (3.10) is proved.

We can now rewrite (3.9) in a more explicit form, with all relevant indices

[Aµ] = detδFα

(x,[A�

µ

])δ�β(y)

∣∣∣∣�=0

; Fα(x, [Aµ]) = 0 (3.11)

Here A�µ stands for Ag

µ and α, β are the gauge group indices. We have to calculatea determinant of a matrix in both space-time and the group indices Mαβ(x, y).This matrix appears in the expansion of Fα

(x,[A�

µ

])in powers of the infinitesimal

parameters �β(y)

Fα(x,[A�

µ

]) = Fα(x, [Aµ])+∫

d4 y Mαβ(x, y)�β(y)+ · · · (3.12)

so that

Mαβ(x, y) = δFα(x,[A�

µ

])δ�β(y)

∣∣∣∣�=0

(3.13)

and

[Aµ] = det M; F[Aµ] = 0 (3.14)

Recalling (1.105) for the infinitesimal transformation of the gauge field, we canalso write

Mαβ(x, y) =∫

d4zδFα(x, [Aµ])

δAρν (z)

δAρν (z)

δ�β(y)

∣∣∣∣θ=0

=∫

d4zδFα(x, [Aµ])

δAρν (z)

(cρβγ Aγ

ν (z)+1

gδρβ∂

(z)ν

)δ(z − y) (3.15)

For the subsequent applications the following observations will be helpful. First


let us note that [Aµ] under a path integral like (3.7) can be replaced by det M ,because the condition F[Aµ] = 0 is already enforced by the δ-functional whichfixes the gauge. Consider a class of gauge conditions of the form F[Aµ]−C(x) =0, where C(x) is an arbitrary function of a space-time point. For all gaugeconditions belonging to this class det M is the same because C(x) is unaffected bythe gauge transformation from Aµ to Ag

µ. We can make use of this feature in orderto replace the δ-functional in (3.7) by some other functional of the gauge condition,which may be more convenient for practical calculations. In the F[Aµ]−C(x) = 0gauge (3.7) becomes(∫

Dg

)∫DAµ det M δ[F[Aµ] − C(x)] f (Aµ) exp{iS[Aµ]} (3.16)

Being gauge-invariant, this is obviously independent of C(x). We can integrate(3.16) functionally over

∫DC with an arbitrary weight functional G[C]; the result

will differ from (3.16) only by an overall normalization constant. Observing thatthe δ-functional in (3.16) is the only C-dependent term, we obtain∫

DAµ det M f (Aµ) exp{iS[Aµ]}∫

DC δ[F[Aµ] − C] G[C]

=∫

DAµ det M f (Aµ) exp{iS[Aµ]}G[F[Aµ]] (3.17)

A popular choice for G[C] is

G[C] = exp

{− i

2α

∫d4x [C(x)]2

}(3.18)

where α is a real constant. This is equivalent to replacing the original lagrangiandensity by

Leff = L− 1

2α(F[Aµ])2 (3.19)

which is no longer gauge-invariant; consequently, perturbation theory may applyin this case. Notice also that in the limit α → 0 (3.18) approaches the δ-functional(up to the α-dependent normalization factor which for our purposes is irrelevant).The derivation of Feynman rules in gauges fixed by δ-functionals can often besimplified by taking this limit.

We have shown how a path integral representation, (3.1), of a gauge-invariantquantity can be rewritten in a fixed gauge, so that the perturbation theory may beset up. The rules of this perturbation theory will depend on the gauge, but the finalresults will not, because the quantity we calculate is gauge-invariant by assumption.In the following we shall also deal with gauge-dependent quantities. In particularthe generating functional W [J ] in the F[Aµ] = 0 gauge is given by the following


expression:

W [J ] = N∫

DAµ [Aµ] δ(F[Aµ]) exp

[i∫

d4x(L+ J α

µ Aµα

)](3.20)

Due to the presence of fixed external sources J αµ , this is not a gauge-invariant object

and the same applies to its functional derivatives with respect to J , the Green’sfunctions of the gauge field. Although in a sense unphysical, the Green’s functionsare often useful at intermediate stages of calculations and physical quantitiesmay be conveniently defined in terms of them. In particular, as we know fromSection 2.7, an element of the S-matrix is obtained from the corresponding Green’sfunctions by removing single-particle propagators corresponding to external lines,taking the Fourier transform of the resulting ‘amputated’ Green’s function andplacing external momenta on the mass-shell. Although we have not discussed therenormalization programme yet, it is in order to mention here that the renormalizedS-matrix elements are gauge-independent. The equivalence of different gauges canbe explicitly checked for various cases (see, for example, Abers & Lee (1973),Slavnov & Faddeev (1980), ’t Hooft & Veltman (1973)). One can show thatonly the renormalization constants attached to each external line depend on thegauge but that is of no consequence for the renormalized S-matrix. The othergauge-dependent terms do not contribute to the poles of the Green’s functionsat p2

i = m2. These formal arguments are rigorous except for the fact that in agauge theory, if not with a spontaneously broken gauge symmetry, the S-matrixsuffers from IR divergences and may not even be defined. In such cases one candiscuss IR regularized theory but the regulator must not break gauge invariance.IR dimensional regularization can be used as an intermediate step to prove gaugeindependence of the physical, finite quantities.

Examples

As a simple example let us consider the case of quantum electrodynamics (QED).We shall calculate the free photon propagators Gµν

0 (x1, x2) corresponding todifferent gauge conditions. First we assume the so-called Lorentz gauge

∂µ Aµ = 0 (3.21)

The M(x, y)-matrix defined by (3.13) is then of the form

M(x, y) = +1

e∂2δ(x − y)

where ∂2 = ∂µ∂µ and the Faddeev–Popov determinant is independent of the field

Aµ; consequently it can be included into the overall normalization factor, which isirrelevant. To deal with the δ-functional δ

[∂µ Aµ

]which fixes the gauge we shall


make use of a decomposition of the field Aµ into transverse ATµ and longitudinal

ALµ parts

ATµ = Pµν Aν, AL

µ = (gµν − Pµν)Aν

where Pµν = gµν − ∂µ∂ν/∂2 is the projection operator. Clearly we have

∂µ Aµ

T = 0

so that the gauge-fixing δ-functional does not affect the integration over thetransverse component of the gauge field. The longitudinal fields are of the formAL

µ = ∂µ� (‘pure gauge’). The Lorentz gauge condition does not set them exactlyto zero; instead, it requires that ∂2� = 0. This is the residual gauge freedomallowed by (3.21).

The existence of this residual gauge freedom contradicts our previous assump-tion that the gauge condition F

[Ag

µ

] = 0 has exactly one solution g0 for any Aµ.However, let us observe that the residual gauge freedom can be removed by anappropriate choice of the boundary conditions. In Euclidean space we can do itby assuming that the gauge field Aµ vanishes at Euclidean infinity. Such boundaryconditions are in fact implicit in the usual derivation of perturbation theory rulesfrom path integrals. There is also another sort of gauge ambiguity (the Gribovambiguity) with more physical implications. In the case of the Lorentz gauge itappears only in non-abelian gauge theories. This is the case when the points ofintersection of a group orbit with the F = 0 surface are at a finite distance fromeach other, i.e. are separated by finite gauge transformations. Perturbation theoryis, however, unaffected by this ambiguity.

Taking this all into account we obtain the following path integral formula forW0[J ] in the Lorentz gauge (in the following the fermion degrees of freedom arealways suppressed):

W0[J ] ∼∫

DATµ exp

[i∫

d4x(

12 AT

µ∂2gµν AT

ν + J Tµ Aµ

T

)](3.22)

(the extra term 12 iεAT

µgµν ATν in the action should always be kept in mind). Here

we have made use of the fact that the lagrangian density of a free electromagneticfield can be written as follows

− 14 Fµν Fµν = 1

2 Aµ∂2(gµν − ∂µ∂ν/∂2

)Aν = 1

2 ATµ∂

2gµν ATν (3.23)

Thus the longitudinal Aµ

L does not appear in the action and in the Lorentz gaugeδ(∂µ Aµ

L

)makes the integration over DAL trivial. We can simply set Aµ

L = 0. Weknow that the J -dependence of a path integral like (3.22) is given by

W0[J ] = exp

[12 i∫

d4x d4 y J Tµ (x) Dµν(x − y) J T

ν (y)

](3.24)


where −Dµν(x − y) is the inverse of the quadratic action. In transverse space theunit operator is equivalent to Pµνδ(x − y), so that we have the following equationfor Dµν : (

∂2 − iε)Dµν(x − y) = −(gµν − ∂µ∂ν/∂

2)δ(x − y) (3.25)

After a Fourier transformation this becomes(−k2 − iε)Dµν(k) = −(gµν − kµkν/k2

)so that

Dµν(k) = 1

k2 + iε

(gµν − kµkν

k2

)= Pµν

k2 + iε(3.26)

and

Dµν(x − y) =∫

d4k

(2π)4

gµν − kµkν/k2

k2 + iεexp[−ik(x − y)] (3.27)

The prescription of the k2 pole in the kµkν/k2 term can be arbitrary. Note also thatin transverse space, as long as we deal only with transverse sources, we can alwaysreplace Pµν by gµν .

We can now write the formula for the free electromagnetic field propagator inthe Lorentz gauge

Gµν

0 (x1, x2) =(

1

i

)2δ2W0[J ]

δ Jµ(x1)δ Jν(x2)

∣∣∣∣J≡0

= −iDµν(x1 − x2) (3.28)

where Dµν(x1 − x2) is given by (3.27).A generalization of this result follows if we make use of another procedure for

fixing the gauge, summarized in (3.17)–(3.19). With F[Aµ] = ∂µ Aµ we obtain theeffective lagrangian

Leff = − 14 Fµν Fµν − 1

2α

(∂µ Aµ

)2 = 12 Aµ

[gµν∂

2 −(

1 − 1

α

)∂µ∂ν

]Aν (3.29)

The quadratic action is now no longer proportional to a projection operator andtherefore can be inverted over all configuration space. Again we can write

W0[J ] = exp

[12 i∫

d4x d4 y Jµ(x) Dµν(x − y) J ν(y)

](3.30)

where Dµν(x − y) satisfies the equation[∂2gµν − ∂µ∂ν(1 − 1/α)− iε

]Dν

λ(x − y) = −gµλδ(x − y) (3.31)


with the solution

Dµν(x − y) =∫

d4k

(2π)4

[(gµν − kµkν

k2 + iε

)+ α

kµkν

k2 + iε

]1

k2 + iεexp[−ik(x − y)]

(3.32)In the limit α → 0 this coincides with (3.27) (the Lorentz or Landau gauge). Thecase α = 1 corresponds to the Feynman gauge.

The free propagators for non-abelian gauge fields in covariant gauges are alsogiven by (3.28) and (3.32). The only modification is that they are diagonal matricesin the non-abelian quantum number space.

Non-covariant gauges

Non-covariant gauges, depending on an arbitrary four-vector nµ, are also fre-quently used. For instance the standard canonical quantization of QED uses theCoulomb gauge, also called the radiation gauge (Bjorken & Drell 1965), definedby the condition

∂µ Aµ − (nµ∂

µ)(

nµ Aµ) = 0, nµ = (1, 0, 0, 0) (3.33)

In QCD one often uses axial gauges

nµ Aµα = 0, n2 = 0 or n2 < 0, α = 1, . . . , 8 (3.34)

which allow the quanta of the vector fields to be interpreted as partons. It is easy tocheck that in axial gauges the Faddeev–Popov determinant is Aα

µ independent evenin a non-abelian theory.

Free propagators of the gauge field in the Coulomb and axial gauges can beformally obtained following the procedure summarized in (3.17)–(3.19), invertingthe term of the action quadratic in the gauge fields and taking the limit α → 0.Only in this limit do the propagators behave as k−2. For the Coulomb gauge onegets

Dαβµν =

δαβ

k2 + iε

[gµν − k · n(kµnν + kνnµ)− kµkν

(k · n)2 − k2

](3.35)

and for axial gauges

Dαβµν =

δαβ

k2 + iε

[gµν − kµnν + kνnµ

k · n+ n2

(k · n)2kµkν

](3.36)

Propagator (3.35) can be split into the transverse propagator and the static Coulombinteraction term (Bjorken & Drell 1965). Our formal derivation of (3.36) neglectssubtle problems raised by the presence of the (n · k) singularity. A more completetreatment of n2 < 0 and n2 = 0 gauges has been given in the framework of

3.2 Feynman rules for QCD 133

canonical quantization (Bassetto, Lazzizzera & Soldati 1985, Bassetto, Dalbosco,Lazzizzera & Soldati 1985). The (n · k) singularities are related to the residualgauge freedom still present after the axial gauge condition is imposed. In thespace-like case n2 < 0 one can show (Bassetto, Lazzizzera & Soldati 1985)that this residual gauge symmetry is directly connected to the spatial asymptoticbehaviour of the potentials and can be eliminated by imposing a specific boundarycondition. Choice of this condition gives rise to a specific prescription for thespurious singularity in the free boson propagator.

The treatment of the light-cone gauge n2 = 0 is quite different (Bassettoet al. 1985). This time the residual gauge freedom involving functions whichdo not depend on the variable nµxµ cannot be eliminated: we are not allowedto impose boundary conditions which would interfere with the time evolutionof the system. Nevertheless a hamiltonian canonical quantization leads to thewell-defined prescription for the singularity (n · k):

1

n · k= k0 + k3

k2 + k2⊥ + iε

, n = (1, 0, 0, 1) (3.37)

This prescription has been proposed earlier without any canonical justificationas one which allows for Wick rotation (Mandelstam 1983, Leibbrandt 1984). InQCD calculations the principal value prescription has also been extensively used.With both kinds of prescription dimensional regularization is usually adopted(Pritchard & Stirling 1979 and Appendix B). However, the general proof of therenormalizability of the theory in the light-cone gauge is still lacking.

3.2 Feynman rules for QCD

Calculation of the Faddeev–Popov determinant

In order to derive Feynman rules for a non-abelian gauge theory we have totake proper account of the Faddeev–Popov determinant whose presence underthe functional integral (3.20) is a consequence of quantization with constraints:gauge-fixing conditions. For the sake of definiteness we choose a class of covariantgauges ∂µ Aµ = c(x) or, equivalently, assume (3.19). It then follows from (3.15)that the Faddeev–Popov matrix reads

Mαβ(x, y) = (1/g)(δαβ∂2 + gcαβγ Aµ

γ (x)∂µ)δ(x − y) = (1/g)

(Dµ∂

µ)αβ

δ(x − y)

(3.38)

Relation (3.38) follows from (3.15), not only on the hypersurface ∂µ Aµ = 0but also for ∂µ Aµ = c(x) because det

(∂µDµ

) = det(Dµ∂

µ). The matrix Mαβ ,

depends on gauge fields Aγµ.


The standard method of dealing with the Faddeev–Popov determinantdet Mαβ(x, y) is to replace it by an additional functional integration over someauxiliary complex fields η(x) (ghost fields) which are Grassmann variables. Usingthe results of Section 2.5 we can write

det Mαβ(x, y) = C∫

DηDη∗ exp

[i∫

d4x d4 y η∗α(x) Mαβ(x, y) ηβ(y)

](3.39)

where C is some unimportant constant (the factor (1/g) in relation (3.38) is alsoincluded in it) and the integration over η and η∗ is equivalent to integration overthe real and the imaginary parts of the field η. Having (3.39) and the explicitform (3.38) of the matrix Mαβ(x, y) we can take account of det Mαβ(x, y) inthe framework of our general perturbative strategy for calculating various Green’sfunctions. The Feynman rules for ghost fields follow in a straightforward way from(3.39) and (3.38). The ghost field propagator can be calculated from the generatingfunctional W η

0 [β]

W η

0 [β] ∼∫

DηDη∗ exp

{i∫

d4x[η∗α(x)δαβ∂2ηβ(x)

+β∗α(x)ηα(x)+ η∗α(x)βα(x)]}

(3.40)

Using standard methods, the β-dependence of W η

0 [β] reads explicitly

W η

0 [β] = exp

[i∫

d4x d4 y β∗α(x) αβ(x − y) ββ(y)

](3.41)

where

αβ(x − y) = δαβ∫

d4k

(2π)4

1

k2 + iεexp[−ik(x − y)] (3.42)

One can define the ‘propagator’ of the ghost field η

Gηαβ

0 (x, y) =(

1

i

)2δ

δβ∗α(x)W η

0 [β]←δ

δββ(y)

∣∣∣∣β=β∗=0

(3.43)

and expanding (3.41) we get

Gηαβ

0 (x, y) = −i αβ(x − y) (3.44)

Graphically, propagator Gηαβ

0 (x, y) will be denoted by

with an arrow pointing towards the source of η.


Feynman rules

The full generating functional W [J, α, β] for QCD reads as follows:

W [J, α, β] = N∫

DAαµ D� D� DηDη∗ exp

{i[S[A, �] + Sη[A, η] + SG]

+ i∫

d4x[J αµ (x) Aµ

α (x)+ β∗α(x) ηα(x)+ η∗α(x) βα(x)

+ α(x)�(x)+ �(x) α(x)]}

(3.45)

where

S[A, �] =∫

d4x(− 1

4 GαµνGµν

α + �iγµDµ� − m��)

Sη[A, η] =∫

d4x η∗α(x)[δαβ∂2 + gcαβγ Aµ

γ (x)∂µ]ηβ(x)

SG = −(1/2α)∫

d4x[∂µ Aα

µ

]2

(3.46)

Dirac spinors �, � describe quarks and are vectors in the three-dimensional colourspace. We split the action into the term S0 describing non-interacting theory(bilinear in fields and including SG)

S0[A, �, η] =∫

d4x[

12 Aα

µ(x)δαβ∂2

(gµν − ∂µ∂ν/∂2

)Aβ

ν (x)

+ �(x)(iγµ∂

µ − m)�(x)+ η∗α(x)δαβ∂2ηβ(x)

]+ SG (3.47)

and into the interaction term SI

SI[A, �, η] =∫

d4x[gcαβγ

(∂µ Aα

ν

)Aµβ Aνγ − 1

4 g2cαβγ cαρσ Aβµ Aγ

ν Aµρ Aνσ

− g�γµ AµαT α� + gcαβγ Aγµη

∗α(∂µηβ)]

(3.48)

Matrices T α are SU (3) colour generators in the fundamental representation. The‘free’ generating functional reads

W0 = exp

{i∫

d4x d4 y[

12 J α

µ (x)Dµναβ (x − y)J β

ν (y)+ α(x)αSαβ(x − y)αβ(y)

+β∗α(x) αβ(x − y)ββ(y)]}

(3.49)

where the Green’s functions Dµναβ , Sαβ and αβ are given by (3.32), (2.122) and

(3.42), respectively. The propagators 〈0|T Aαµ(x)Aβ

ν (y)|0〉, 〈0|T�α(x)�β(y)|0〉and 〈0|Tηα(x)η∗β(y)|0〉 are obtained by multiplying the respective D-, S- and -functions by (−i).


Fig. 3.2.

Any Green’s function in QCD can be calculated perturbatively from the expan-sion

〈0|T Aαµ . . . Aβ

ν� . . . �|0〉

=

∫D[A��ηη∗]

(Aα

µ . . . Aβν� . . . �

)∑N

(iSI)N

N !exp(iS0)∫

D[A��ηη∗]∑

N

(iSI)N

N !exp(iS0)

(3.50)

We recall that the presence of the denominator in (3.50) eliminates all vacuum-to-vacuum diagrams. One is usually interested in Green’s functions in momentumspace.

We shall now consider several Green’s functions in the first order in SI to deriveFeynman rules for vertices in QCD. We begin with

G(3g)abcµνλ (x1, x2, x3) ≡ 〈0|T Aa

µ(x1)Abν(x2)Ac

λ(x3)|0〉 (3.51)


which defines the full symmetrized triple gluon (3g) vertex by means of the relation

G(3g)abcµνλ (x1, x2, x3) =

∫d4 y

[−iDaa′µµ′(x1 − y)

][−iDbb′νν′ (x2 − y)

]× [−iDcc′

λλ′(x3 − y)]V µ′ν′λ′

a′b′c′ (y) (3.52)

illustrated in Fig. 3.2. In the first order in SI one has

G(3g)abcµνλ (x1, x2, x3) = 1

i6δ3

δ J aµ(x1)δ J b

ν (x2)δ J cλ (x3)

i∫

d4 y(gcdef ∂

(d)ρ

)× δ3

δ J dκ (y)δ J e

ρ (y)δ J f κ(y)W0[J, β, α]|J=β=α=0 (3.53)

(the superscript ‘d’ in ∂(d)ρ means that the component Ad

κ is differentiated withrespect to y) and the cubic term in the expansion of W0 contributes to the finalresult. Following the standard method we assign points x1, x2, x3 and y1, y2, y3 →y to the prototype diagram

For a connected diagram there are six different orderings in pairs joined bypropagators. For each assignment there is a 233! symmetry factor which cancelswith factors coming from the expansion of W0. The factor

(1/i6

)i3 = (−i)3 is

included in the three propagators. The final result reads:

G(3g)abcµνλ (x1, x2, x3) = igcdef

×∫

d4 y ∂(d)ρ

∑[−iDadµκ(x1 − y)

][−iDbeνρ(x2 − y)

][−iDcf κλ (x3 − y)

](3.54)

where the sum denotes all permutations of sets of indices (d, κ), (e, ρ) and ( f, κ)and the derivative is with respect to y. Writing the result (3.54) in the form (3.52)we get

V abcµνλ = igcdef

(δadδbeδc f gµλ∂

(d)ν + δa f δbdδcegµν∂

(d)λ + δaeδb f δcd gνλ∂

(d)µ

+ δadδb f δcegµν∂(d)λ + δaeδbdδc f gνλ∂

(d)µ + δa f δbeδcd gµλ∂

(d)ν

)(3.55)

In momentum space we define G(3g)abcµνλ (p, q, r) as

G(3g)abcµνλ (x1, x2, x3) =

∫d4 p

(2π)4

d4q

(2π)4

d4r

(2π)4exp(−ip · x1) exp(−iq · x2)

× exp(−ir · x3) G(3g)abcµνλ (p, q, r) (3.56)


and the 3g vertex V µνλ

abc (p, q, r) by the relation

G(3g)abcµνλ (p, q, r) = G(2g)aa′

µµ′ (p)G(2g)bb′νν′ (q)G(2g)cc′

λλ′ (r)V µ′ν′λ′a′b′c′ (p, q, r) (3.57)

where

G(2g)aa′µµ′ (p) = −i

p2 + iε

[gµµ′ − (1 − α)

pµ pµ′

p2

]δaa′

etc. Writing the propagators in relation (3.52) in terms of their Fourier transforms(3.57), differentiating according to (3.55), integrating over d4 y and finally compar-ing with (3.56) and (3.57) one gets the final result

V abcµνλ(p, q, r) = −(2π)4δ(p + q + r)gcabc

× [(r − p)νgµλ + (q − p)λgµν + (r − q)µgνλ] (3.58)

(where all momenta are outgoing). Similarly, one can derive the expression for the4g vertex. The starting point is the Green’s function

G(4g)abcdµνλσ (x1, x2, x3, x4) ≡ 〈0|T Aa

µ(x1)Abν(x2)Ac

λ(x3)Adσ (x4)|0〉

which in the first order in SI defines the 4g vertex according to the followingrelation:

G(4g)abcdµνλσ (x1, x2, x3, x4) = (−i)4

∫d4 y Daa′

µµ′(x1 − y) Dbb′νν′ (x2 − y)

×Dcc′λλ′(x3 − y) Ddd ′

σσ ′(x4 − y) V µ′ν′λ′σ ′a′b′c′d ′ (3.59)

Writing

G(4g)abcdµνλσ (x1, x2, x3, x4) = 1

i8δ4

δ J aµ(x1)δ J b

ν (x2)δ J cλ (x3)δ J d

σ (x4)

× i∫

d4 y(− 1

4 g2cef hcemn) δ4

δ J fκ δ J h

ρ δ J mκδ J nρ(y)W0|J=β=α=0 (3.60)

one can calculate V abcdµνλσ . There are 24 different assignments of pairs of points

x1 . . . x4, y1 . . . y4 to the prototype diagram. The additional 4!24 symmetry factoris cancelled by the same factor coming from the expansion of W0. Rememberingthat the Lorentz contraction in the 4g vertex always occurs between colour indicesf and m and h and n one gets the final result:

V abcdµνλσ = −ig2[cabeccde(gµλgνσ − gµσ gνλ)+ cacecbde(gµνgλσ − gµσ gνλ)

+ cadecbce(gµνgλσ − gµλgνσ )] (3.61)

which is obviously also true in momentum space.


Fig. 3.3.

Fig. 3.4.

Following the functional approach one can also easily find that loop diagramsin Fig. 3.3 contribute with additional factors 1

2 , 12 and 1

6 , respectively. Inparticular diagrams (a) and (b) occur in the second order in SI; e.g. the expressioncorresponding to the first diagram is of the form

(iSI)2

2!

1

4!

1

24J D J J D J J D J J D J (3.62)

The number of different assignments to the prototype diagram is 3 × 2 × 3 × 2 ×4!24. Noticing that the product of two vertices gives us according to (3.58) 36combinations we are left with an additional factor 1

2 . In a similar way one canobtain the other factors.

The final task is to find expressions for the ghost and fermion vertices in Fig. 3.4.They can be obtained from the Green’s functions 〈0|Tηc(x3)η

b �(x2)Aaµ(x1)|0〉 and

〈0|T� j (x3)�i (x2)Aaµ(x1)|0〉, respectively, written as

∫d4 y

[−iDaa′µµ′(x1 − y)

][−i bb′(x2 − y)][−i cc′(x3 − y)

]V µ′

a′b′c′(y) (3.63)

and similarly for the ��A vertex. Following the previous calculations closely we


obtain in momentum space

Diagram 3.4(a): V abcµ = −gccabrµ(2π)4δ(r + q − p) (3.64)

Diagram 3.4(b): V abcµ = −igγµ

(T a)

j i(2π)4δ(r + q − p) (3.65)

Of course, Feynman rules for QED in the covariant gauge can be derivedanalogously. All Feynman rules are given in Appendices B and C.

3.3 Unitarity, ghosts, Becchi–Rouet–Stora transformation

Unitarity and ghosts

A spin-one massless particle has only two physical polarization states εµ(λ, k),λ = 1, 2. Since the three four-vectors kµ, εµ(λ, k) do not span four-dimensionalspace, the usual on-shell Lorentz condition k · ε = 0 for spin-one massive particlesis not enough to determine the photon polarization vectors uniquely. To make themunique we must sacrifice manifest Lorentz covariance and, for instance, chooseanother vector nµ such that

k · n �= 0, n · ε = 0 (3.66)

Together with equations

k · ε = 0, ε∗(λ1, k)ε(λ2, k) = −δλ1λ2 (3.67)

conditions (3.66) specify the vectors ε(λ, k) uniquely. The polarization sum reads∑λ=1,2

ε∗µ(λ, k)εν(λ, k) = −gµν + kµnν + kνnµ

n · k− n2kµkν

(k · n)2≡ Pµν (3.68)

In field theory every particle is associated with a field with definite transfor-mation properties under the homogeneous Lorentz group. Thus the four-vectorgauge field Aµ(x) actually represents only two physical degrees of freedom. Animportant point is that fixing the gauge is usually not sufficient to totally removethe unphysical degrees of freedom from the theory (an exception is the axialgauge with n2 < 0). A well-known example of their presence in the theory isthe Gupta–Bleuler formulation of QED in the covariant gauge ∂µ Aµ = 0. Ingeneral, such degrees of freedom can be seen explicitly if we construct the Fockspace of the theory: a longitudinal photon state and a negative norm scalar photonstate are present. A physically sensible theory can be defined in Hilbert spacerestricted to the transverse photons (see, for example, de Rafael (1979)). However,this restriction does not prevent one from having unphysical states propagating asvirtual states in the intermediate steps of perturbative calculations. Indeed, the

3.3 Unitarity, ghosts, BRS transformation 141

Fig. 3.5.

gauge boson propagators are gauge-dependent. The question then arises about theunitarity of the theory.

The requirement that the S-matrix is unitary implies that the scattering amplitudeT , defined as

Sif = δif + i(2π)4δ(pi − pf)Tif (3.69)

satisfies the relation

Tif − T ∗fi = i

∑n

TinT ∗fn(2π)4δ(pi − pn) (3.70)

where the sum is taken over all physical states for which transitions to the initialand final states are allowed by quantum number conservation laws. For the forwardelastic scattering the l.h.s. of (3.70) is just the imaginary part of the amplitude Tii

and in the general case it is the absorptive part of Tif (see, for example, Bjorken &Drell (1965), Section 18.12). The absorptive part of the scattering amplitude canbe calculated perturbatively by means of the so-called Landau–Cutkosky rule: thecontribution due to a Feynman diagram with a given intermediate state, Fig. 3.5,is obtained from the Feynman amplitude by replacing the propagators in thisintermediate state by their imaginary parts. For instance, for the gauge bosonpropagators in the Feynman gauge (see Section 3.1)

Dαβµν = δαβ

gµν

k2 + iε→−2π iδαβgµνδ

(k2)�(k0) (3.71)

It is clear that in gauges which do not remove the unphysical degrees of freedom,like in the Feynman gauge, the unphysical intermediate states are included in ourcalculation. On the other hand we can calculate the absorptive part of the scatteringamplitude from the r.h.s. of the unitarity relation (3.70). Here we sum over physicalstates only. Thus the unitarity condition demands that the unphysical intermediatestates do not contribute to the amplitude when also calculated by means of theLandau–Cutkosky rule. In fact one can expect that gauge invariance, which reflectsthe presence of the redundant degrees of freedom, ensures at the same time theirdecoupling. An explicit verification of this point may, however, be quite subtle as


Fig. 3.6.

for solving the equations of motion we need a gauge-fixing term which breaks themanifest invariance. Nevertheless in QED there is no problem due to the fact thatphotons always couple to conserved currents

kµMµν = 0 (3.72)

Therefore Pµν given by (3.68) can be replaced by

Pµν →−gµν + αkµkν (3.73)

where α is an arbitrary quantity.In non-abelian gauge theories, the situation is more complicated as can be

illustrated by the standard example of the fermion–antifermion scattering in thelowest non-trivial order in the gauge-coupling constant (Feynman 1977, Aitchison& Hey 1982). The sum of the diagrams with only gauge particles in the inter-mediate state (Fig. 3.6) calculated in a covariant gauge (3.19) does not satisfythe unitarity condition (3.70), where the r.h.s. amplitudes Tin and T ∗

fn are givenby the diagrams in Fig. 3.7 with only physical gluons as the state n (Problem 3.1).However, we have learned in Sections 3.1 and 3.2 that a consistent quantization ofa non-abelian gauge theory requires the addition to the lagrangian of an extra term,the so-called ghost term (3.39). It turns out that these additional unphysical degreesof freedom cancel the gauge field unphysical polarization states exactly, leading toa perfectly unitary theory. In our specific example the cancellation occurs dueto the contribution of the diagram with the ghost loop, shown in Fig. 3.8. Anexplicit perturbative calculation (Problem 3.1) shows that certain relations betweenthe f f → Aα

µ Aβν and f f → ηαηβ amplitudes, where Aµ and η are the gauge and

ghost fields, respectively, are responsible for the cancellation. These are examplesof Ward identities for a non-abelian gauge theory.


Fig. 3.7.

Fig. 3.8.

BRS and anti-BRS symmetry

We have introduced ghosts as a technical device to express the modification ofthe functional integral measure required to compensate for gauge-dependenceintroduced by the gauge-fixing term. The discussion in the previous subsectionsuggests, however, that it is very natural to adopt the point of view that ghosts, i.e.fields with unphysical statistics, are necessary to compensate for effects due to thequantum propagation of the unphysical states of the gauge fields. Provided oneis able to express the gauge symmetry in a form involving both the classical andghost fields one can introduce both fields from the very beginning as fundamentalfields of a gauge theory and construct its lagrangian requiring invariance underthis extended symmetry principle. Such a symmetry, the so-called BRS andanti-BRS symmetry, has indeed been discovered, originally as a symmetry ofthe Faddeev–Popov lagrangians with specifically chosen gauge-fixing conditions.We shall first follow this historical path and later comment on the extensionwhich is independent of the notion of a lagrangian and remains in the spirit ofthe above outlined programme. It turns out that given any gauge symmetry onecan always find symmetry transformations acting on gauge and ghost fields suchthat invariance under this symmetry is equivalent to gauge invariance of physicalquantities.

To introduce BRS symmetry (Becchi, Rouet & Stora 1974, 1976) let us considerthe action including the gauge-fixing term and the ghost term

Seff = S +∫

d4x

[− 1

2α

(Fα[x, A(x)]

)2 + η∗Mη

]= S + SG + Sη (3.74)


where

(Mη)α =∫

d4 y Mαβ(x, y) ηβ(y)

and

Mαβ(x, y) =∫

d4zδFα[x, A(x)]

δAρµ(z)

Dρβµ (z)δ(z − y) (3.75)

and

Dρβµ = ∂µδ

ρβ + gcρβγ Aγµ

Thus

(Mη)α = δFα[x, A(x)]

where δFα is the variation of the functional Fα under a gauge transformation with�β(x) replaced by

(gηβ(x)

). The full action (3.74) is invariant under the global

transformation

δBRS Aαµ = Dαβ

µ �ηβ ≡ �(s Aα

µ

)δBRS� = −igT α�ηα� ≡ �(s�)

δBRSηα = + 1

2 gcαβγ ηβηγ� ≡ �(sηα

)δBRSη

∗α = +(1/α)�Fα ≡ �(sη∗α

)

(3.76)

where � is a space-time-independent infinitesimal Grassmann parameter (notethat it is not necessary to think of η and η∗ in (3.39) as hermitean conjugates:see Problem 3.4). We observe that for the classical fields Aµ and � the BRStransformation has the form of a gauge transformation with

�α(x) = +g�ηα(x) (3.77)

Therefore the BRS transformation leaves the original action invariant without thegauge-fixing and the ghost terms. Thus it remains to show that

δBRS(SG + Sη) = 0 (3.78)

This can be checked by a straightforward calculation. As intermediate steps werecord the relation

δBRS(Mη) = 0 (3.79)

and the fact that the transformations on Aαµ, � and η are nilpotent (see Problem 3.2)

s2 Aαµ = s2� = s2ηα = 0 (3.80)

(but s2η∗α = (Mη)α). We note also that the measure D(A�ηη∗) is invariant underthe BRS transformations.


The BRS symmetric lagrangian can also be written down in QED. In this casethe ghost field is a free real scalar field and the covariant derivative in the firstequation in (3.76) is replaced by the normal derivative (Problem 3.3).

The BRS transformation can be written down in a form more symmetric withrespect to the fields η and η∗ if we introduce an auxiliary field b. This also allowsformulation of the notion of BRS symmetry in a lagrangian-independent way(transformations (3.76) depend on the gauge-fixing term of the lagrangian). Thisstep is very important for promoting BRS symmetry to the fundamental symmetryof gauge theories. To be specific let us discuss the case of a covariant gauge. Usingthe freedom we have to choose any gauge condition without affecting the S-matrixwe can take

LG = bα∂µ Aµα + 1

2αbαbα (3.81)

as our gauge-fixing condition. After taking into account the equation of motionfor the field b we see that (3.81) is equivalent to the standard gauge fixing term−(1/2α)

(∂µ Aα

µ

)2. Using the auxiliary field b the BRS transformation for η∗ and b

reads

sη∗α = bα, sbα = 0 (3.82)

The action of s on an arbitrary function of the fields follows from their action onfield polynomials

s(AB) = (s A)B ± As B (3.83)

where the minus sign occurs if there is an odd number of ghosts and antighosts inA. The nilpotency of BRS symmetry can now be expressed as

s2 = 0 (3.84)

It has also been discovered (Curci & Ferrari 1976, Ojima 1980) that in additionto BRS symmetry there is another symmetry which leaves the quantum Yang–Millsaction (3.74) invariant provided the gauge-fixing functional Fα is linear in fields.These anti-BRS transformations read

s Aαµ = Dαβ

µ η∗β

sη∗α = 12 gcαβγ η∗βη∗ γ

sηα = −bα + gcαβγ η∗βηγ

sbα = gcαβγ η∗βbγ

s� = −igT αη∗α�

(3.85)

and

ss + ss = s2 = 0 (3.86)


We have introduced the BRS and anti-BRS transformations as symmetry trans-formations of a Yang–Mills lagrangian with a gauge-fixing term linear in fields.With the auxiliary field b interpreted as a Lagrange multiplier for the gauge-fixingcondition, the symmetry generators s and s become independent of the notion ofthe lagrangian and satisfy the nilpotency relations (3.84) and (3.86). In terms ofthe fields, which are matrix-valued in the Lie algebra of the gauge group, we have

s Aµ = Dµη s Aµ = Dµη∗

s� = −η� s� = −η∗�

sη = − 12 [η, η] sη∗ = − 1

2 [η∗, η∗]

sη∗ = b sη = −[η∗, η] − b

sb = 0 sb = −[η∗, b]

(3.87)

An important next step is to promote full BRS (BRS or/and anti-BRS) symmetryto the fundamental symmetry of a gauge theory (Alvarez-Gaume & Baulieu 1983).In fact, one can show that starting from a set of infinitesimal gauge transformationsbuilding up a closed algebra (possibly with field-dependent structure functions)with a Jacobi identity one can always build an associated nilpotent full BRSalgebra. One can also prove that full BRS invariance of the lagrangian leads tothe gauge independence of physics. Thus we are led to construct the quantumlagrangian of a gauge theory as the most general s- or/and s-invariant andLorentz-invariant function (with ghost number zero) of physical and ghost fields.This construction simultaneously generates gauge-fixing and ghost terms in thelagrangian. It is not just a formal improvement of the Faddeev–Popov approachsince it also gives consistent quantum theories in cases for which the Faddeev–Popov method is inapplicable. Actually, the latter is a consistent approach only forfour-dimensional Yang–Mills theories when one chooses a linear gauge condition.Specifying the general BRS-invariant lagrangian to this special case we indeedrecover the lagrangian of Faddeev and Popov.†

In the general case, in a BRS-invariant lagrangian four-ghost or higher ghost in-teractions are present which cannot be obtained from the ordinary Faddeev–Popovprocedure. Such couplings are in fact necessary for a consistent renormalizableYang–Mills theory if the gauge-fixing functional Fα is chosen to be non-linear infields. Indeed, in such cases, one-loop calculation, based on the Faddeev–Popovlagrangian generating only quadratic ghost interactions, shows that the four-pointghost function is divergent. Thus, four-ghost interaction terms should have beenintroduced at the tree level. Also, in supergravity, quartic ghost interactions, absentin the Faddeev–Popov approach, have been shown to be necessary (Kallosch 1978,

† In Problem 3.5 this is discussed for a less general lagrangian, which is simultaneously BRS and anti-BRSinvariant.

Problems 147

Nielsen 1978, Sterman, Townsend & van Nieuwenhuizen 1978). They are naturallygenerated by determining the quantum lagrangian from the requirement of itsBRS invariance. Thus, full BRS invariance indeed emerges as the fundamentalsymmetry of the quantum theory associated with any given underlying gaugeinvariance.

Problems

3.1 Calculate in a covariant gauge the absorptive part of the amplitude corresponding tothe sum of diagrams in Fig. 3.6 and Fig. 3.8 and check the unitarity condition (3.70).

3.2 Prove the invariance of (3.74) under transformation (3.76). Check (3.79) and (3.80).(Use the relation for the structure constants which follows from the commutationrelations

[T a, T b

] = icabcT c with(T a)

bc = −icabc.)

3.3 Check the invariance of the generalized QED lagrangian

L = − 14 Fµν Fµν + �(iD/ − m)� − (1/2α)

(∂µ Aµ

)2 − 12 (∂µω)

(∂µω

)where ω(x) is a free scalar field, under the transformation

δ� = +ie�ω(x)�(x)

δAµ = �∂µω(x)

∂ω = (1/α)�∂µ Aµ

3.4 Note that the fields η and η∗ in (3.39) can be any anticommuting fields. Check that, inparticular, they can be taken as real and imaginary parts of η, i.e. as hermitean fields.Check that due to the phase arbitrariness of the exponent in (3.39) the Faddeev–Popovterm Lη can be made hermitean.

3.5 Check that the most general Yang–Mills and Lorentz scalar lagrangian which is BRS-and anti-BRS-invariant and has ghost number zero can be written as follows (Baulieu& Thierry Mieg 1982)

L = − 14 (Fµν)

2 − 12 (Dµ�)2 + �iD/� + Yukawa terms + εµνρσ Fµν Fρσ

+ Vinv(�)+ ss( 1

2 A2µ + βη∗η + γ 〈�〉�)+ 1

2αb2

where α, β, γ are arbitrary gauge parameters and 〈�〉 is the vacuum expectation valueof the boson field �. When β = 0 one recovers the usual Faddeev–Popov lagrangianfor linear gauges.

3.6 Use the superspace formulation of the Yang–Mills theory (Bonora & Tonin 1981,Hirshfeld & Leschke 1981) to construct the BRS- and anti-BRS-invariant lagrangians.Follow the same method in supersymmetric Yang–Mills theory (Falck, Hirshfeld &Kubo 1983) and in supergravity.

4

Introduction to the theory of renormalization

4.1 Physical sense of renormalization and its arbitrariness

Bare and ‘physical’ quantities

Let us consider a theory of interacting fields defined by a given lagrangian density;for definiteness, let us take the case of a scalar field with quartic coupling

L = 12(∂µ�B)

2 − 12 m2

B�2B −

λB

4!�4

B (4.1)

We know already how to derive the Feynman rules which allow us to calculate theGreen’s functions of this theory in the form of the perturbative expansion in powersof the coupling constant λB. However, some of the loop momentum integrationsin the resulting Feynman diagrams are divergent in the UV, reflecting the factthat, most likely, our theory does not properly describe the physics at very shortdistances. Consequently, the coefficients of the perturbative series are, in general,infinite. Does this mean that the lagrangian (4.1) cannot be used to define a theory?

In the following we shall show that this conclusion is not necessarily true.Two basic steps in interpreting a theory like (4.1) are the regularization ofdivergent integrals and the subsequent renormalization of the parameters and theGreen’s functions. A regularization procedure is necessary to make sense of theintermediate steps of perturbative calculations. The renormalization procedureaccounts for the fact that, for instance, the ‘bare’ coupling constant λB is nota correct expansion parameter. This is not so unnatural as it seems: the ‘bare’parameters of the theory, like λB and the ‘bare’ mess mB (as well as the Green’sfunctions of the ‘bare’ field �B) are not directly related to observable quantities.

After renormalization, the observable quantities are expressed in terms ofexperimentally measured parameters and our ignorance about the short-distancephysics is hidden in those free parameters of the theory determined by experiment.We shall now discuss this problem in some detail.

148

4.1 Physical sense of renormalization and arbitrariness 149

First let us recall the definition of the ‘physical’ field �F given in Section 2.7:

〈k|�F(x)|0〉 = exp[i(k0x0 − k · x)]; k0 = (k2 + m2F)

1/2 (2.192)

The ‘physical’ field �F is thus defined by a normalization condition on theone-particle state created by �F. The on-shell truncated Green’s functions ofthe physical field have a straightforward physical interpretation: they are equalto the appropriate S-matrix elements. Off-shell the equality also holds for analyticcontinuations.

The normalization of a field operator �(x) is directly related to the on-shellresidue of its two-point Green’s function in momentum space. This is convenientbecause we prefer to deal with Green’s functions and not field operators through-out. Consider thus the two-point Green’s function of the field �

G(2)(x, y) = 〈0|T�(x)�(y)|0〉 (4.2)

and insert the complete set of states (2.189) between the �s. The contribution ofthe one-particle intermediate state is then of the form∫

d3k

(2π)32k0[〈0|�(x)|k〉〈k|�(y)|0〉�(x0 − y0)

+〈0|�(y)|k〉〈k|�(x)|0〉(y0 − x0)]

=∫

d4k

(2π)3�(k0)δ(k

2 − m2F){�(x0 − y0) exp[−ik(x − y)]

+�(y0 − x0) exp[ik(x − y)]}|〈0|�(0)|k〉|2

=∫

d4k

(2π)4

i

k2 − m2F + iε

exp[−ik(x − y)]|〈0|�(0)|k〉|2 (4.3)

(the last step can be easily checked by calculating the r.h.s. first). Therefore, theFourier-transformed two-point function near the mass-shell behaves as

G(2)(k) ≈k2→m2

F

i

k2 − m2F + iε

|〈0|�(0)|k〉|2 (4.4)

note that 〈0|�(0)|k〉 is a constant from relativistic invariance. If �(x) = �F(x) itnow follows from (2.192) that

G(2)F (k) ≈

k2→m2F

i

k2 − m2F + iε

(4.5)

The result (4.5) for the ‘physical’ field �F does not, in general, agree withthe result of the perturbation theory calculation for ‘bare’ Green’s function:G(2)

B (x, y) = 〈0|T�B(x)�B(y)|0〉 (with some regularization introduced to keep

150 4 Introduction to the theory of renormalization

the loop integrals finite). We shall find instead that

G(2)B (k) ≈

k2→m2F

Z3

k2 − m2F + iε

(4.6)

where

Z3 �= 1

and is actually divergent in the absence of regularization. The ‘bare’ field �B isthus not ‘physical’.

We can, however, express the original field �B in terms of the physical field �F

�B = Z1/23 �F (4.7)

which implies a simple relation between the Green’s functions (connected or not):

G(n)F (x1, . . . , xn) = Z−n/2

3 G(n)B (x1, . . . , xn) (4.8)

and its counterpart for the connected proper vertex functions

�(n)F (x1, . . . , xn) = Zn/2

3 �(n)B (x1, . . . , xn) (4.9)

which also holds for the truncated Green’s functions that appear in the expres-sion (2.190) for the S-matrix elements. Consequently, the relation between theS-matrix and ‘bare’ Green’s functions involves the renormalization constant Z3.The ‘bare’ Green’s functions by themselves are not observable. On the otherhand, the ‘physical’ Green’s functions are observable and must come out finite inperturbation theory. This is not yet the case in the ‘bare’ perturbation expansion inλB. Suppose that we define the coupling constant λF as the value of the connectedproper vertex function �

(4)F (pi ) at some specified point, the ‘renormalization point’,

in momentum space. We can then attempt to express other physical Green’sfunctions as perturbation series in powers of λF. In this way we would obtainthe expression of one observable quantity, the physical Green’s function, in termsof other observable quantities: λF and the physical mass. We expect that a relationbetween observables should be free from infinities. Consequently, the coefficientsof the resulting perturbation expansion should be finite for observable quantities ifthe perturbation theory applies.

Let us denote the renormalization point by µ. By definition we have

�(4)F |µ = λF (4.10)

We can use the regularized bare perturbation expansion to calculate �(4)B at the same

point. The result we shall write as

�(4)B |µ = λB

Z1(4.11)


Fig. 4.1.

which defines the new renormalization constant Z1. Using (4.9) we now obtain

λB = (Z1/Z23)λF (4.12)

The physical mass m2F is defined as the square of the four-momentum of a freely

propagating particle (see (2.192)). From the unitarity condition for the S-matrix,expressed in terms of the scattering amplitude Ti f , (3.69) and (3.70), it then followsthat the amplitude Ti f for the process i → f which can proceed via a one-particleintermediate state has a pole at p2 = m2

F. In perturbation theory the generalstructure of the amplitude with a one-particle intermediate state corresponds to thediagram in Fig. 4.1, where the central bubble denotes the full propagator. Thereforethe full propagator must have a pole at p2 = m2

F

G(2)(p) = R

p2 − m2F

+ · · ·

and the physical mass can also be defined as the value of p2 for which the fullpropagator has a pole. Defined as a pole of the scattering amplitude, m2

F is anexperimentally measurable parameter.

Equivalently, we shall define m2F as the position of the zero of the inverse

propagator (see (2.152))

�(2)(p2 = m2F) = 0 (4.13)

which can be either ‘bare’ or ‘physical’. This definition is convenient because itavoids multiple poles present in G(2)(p) calculated to any finite order. Again, wecan calculate m2

F in the regularized bare perturbation expansion and define the massrenormalization constant Z0 by the relation

m2F = (Z3/Z0)m

2B (4.14)

Altogether, we have three independent renormalization constants: Z0, Z1 andZ3 corresponding to three relations, (4.14), (4.12) and (4.7), between ‘bare’ and‘physical’ quantities. Using these relations we can now rewrite the original


lagrangian (4.1) in terms of new fields �F and parameters mF, λF

L = 12(∂µ�B)

2 − 12 m2

B�2B − (λB/4!)�4

B = 12 Z3(∂µ�F)

2 − 12 Z0m2

F�2F

− Z1(λF/4!)�4F = 1

2(∂µ�F)2 − 1

2 m2F�

2F − (λF/4!)�4

F

+ 12(Z3 − 1)(∂µ�F)

2 − 12(Z0 − 1)m2

F�2F − (Z1 − 1)(λF/4!)�4

F (4.15)

where

�B = Z1/23 �F, m2

B = Z0 Z−13 m2

F, λB = Z1 Z−23 λF (4.16)

The resulting ‘renormalized’ lagrangian we shall use as the basis for the formula-tion of the new perturbation expansion parametrized by the ‘physical’ quantities:mF and λF. It is this ‘physical’ perturbation theory and its generalizations describedbelow that will be used in calculations: we shall not refer to the ‘bare’ lagrangianexplicitly.

The final important remark is the following. We have introduced the renormal-ization constants as multiplicative constants. This is not necessary, and for theparameters of the lagrangian (couplings and masses) it is not always convenient. Aphysically important example is, for instance, a scalar field theory with two fields,φ and �, with the masses m and M , respectively. In this case, the renormalizationof m2 requires an additive term δm2 which has a part proportional to M2 and viceversa. Another important example is a theory of scalar and fermion fields, � and�, with the couplings λ�4 and g��) (Yukawa coupling). The former is, forexample, renormalized by the fermion loop contributing to the proper 4� vertexfunction and this contribution is proportional to g4 but not to λ. Thus, an additiverenormalization of λ, λB = Z−2

3 (λF+δλ), is more appropriate. The latter notation,m2

B = Z−13 (m2

F+δm2) and λB = Z−23 (λF+δλ) is, in fact, general enough to include

both cases.

Counterterms and the renormalization conditions

In (4.15) we have divided the lagrangian into two parts. The first part isidentical to (4.1) with the original parameters replaced by physical ones. Theperturbation theory based on this part of the lagrangian alone, unless subject tosome regularization procedure, would give rise to divergences analogous to thoseof the ‘bare’ perturbation expansion. We expect that these divergences will becancelled order-by-order if we take into account the second, Zi -dependent part ofthe lagrangian: the ‘counterterms’. The counterterms can be properly defined onlyin a regularized theory. After the cancellation of the regulator-dependent terms,the regulator can be removed. If this procedure works in any order of perturbationtheory with finite number of counterterms, we say that the theory is renormalizable.


The theory defined by (4.1) is renormalizable if considered in the space-time ofdimension d ≤ 4.

The renormalization constants Z0, Z1 and Z3 are, of course, not observablequantities and even if expressed in the new ‘physical’ perturbation expansion interms of λF and mF are divergent order-by-order after the regularization is removed.

Now let us make the following important observation concerning the practicalapplication of our results. We shall always use the ‘physical’ perturbationexpansion and never refer to the ‘bare’ parameters at any stage of calculations.From this point of view to define Z3, Z1 and Z0 by relations like (4.7), (4.12) and(4.14) is inconvenient because these relations involve ‘bare’ quantities. Instead,we shall fix the counterterms by imposing the ‘renormalization conditions’ on thephysical Green’s functions. We have, in fact, already used one such condition todefine λF: it is (4.10). To fix all counterterms we need two more conditions, whichwe choose as follows

∂

∂p2�

(2)F (p)|p2=m2

F= 1 (4.17)

which implies

�(2)F (p) −→

p2=m2F

p2 − m2F

in accord with the fact that field �F in (4.15) is assumed to be the ‘physical’ field;and

�F(p)|p2=m2F= 0 (4.18)

where

�(2)F (p) = p2 − m2

F −�F(p) (4.19)

This choice is exactly equivalent to our previous definitions. The counterterms,present in the Feynman rules of the ‘physical’ perturbation expansion, can bedetermined from the renormalization conditions order-by-order. Observe that wenow use mF and λF as free parameters rather than mB and λB: this is also morenatural physically.

Arbitrariness of renormalization

We have outlined the procedure which replaces the original ‘bare’ perturbationexpansion with divergent coefficients by a new perturbation expansion in which thecoefficients are finite. How unique is this procedure? We have used the physicalobservability of new parameters and Green’s functions as a guide in order to ensurethat they are finite and in finite relation to each other. However, if the perturbationtheory parametrized by mF and λF is finite we would not expect to lose this property


if we change the parametrization and use parameters which differ from mF and λF

by a finite amount. In other words once we have specified the counterterms whichcancel the infinities we can make a finite change in them. We thus have an infinite,n-parameter class of different ‘renormalization prescriptions’ (n is the number ofcounterterms). This renormalization symmetry of the theory is very important,and not only because it allows one to choose the most convenient renormalizationprescription; it is the basis of the renormalization group and the renormalizationgroup equations.

Consider some examples. First of all we can change the renormalization pointµ:µ → µ′. As a result, the value of λF changes: λF → λ′F. As the bare parametersare fixed, also the renormalization constant Z1, defined by (4.11), must change:Z1 → Z ′

1. It is possible to do this transformation without affecting Z3 or Z0: thefield �F is still given by (4.7) and therefore does not change; however, it will nowhave to be expressed in terms of the new parameter λ′F. The Green’s functions arestill ‘physical’ but have a new perturbation expansion, in powers of λ′F.

So far we have kept to the original, ‘physical’ scheme. Let us now change thevalue of the mass renormalization constant Z0: Z0 → Z ′

0 keeping Z3 unchanged.As m2

B is fixed and the physical content of the theory should not change we mustconclude that the new mass parameter, the ‘renormalized mass’, defined as

m2R = (Z3/Z ′

0)m2B (4.20)

is no longer the physical mass. In terms of the ‘renormalization conditions’discussed before this may correspond to replacing (4.18) by

�R(p)|µ = 0 (4.21)

where now

�(2)R = p2 − m2

R −�R(p) (4.22)

and µ is some renormalization point.Similarly, we do not have to require that the coupling constant parameter is equal

to the value of �(4)F at any renormalization point. We can use any definition as long

as the transition from λF to the new parameter λR is perturbatively finite: i.e. λR

can be expressed in terms of λF in the λF-perturbation expansion, or vice versa.One important example of a renormalization procedure which makes use of thisgeneralization is the minimal subtraction scheme, which we shall discuss in detailin the following sections.

Finally we can relax the requirement of the ‘physical’ renormalization ofthe two-point renormalized Green’s function. Instead of �F we can use the‘renormalized’ field �R provided that their relative normalization is perturbatively


finite, i.e. we have

�R = Z1/2R �F (4.23)

where ZR = ZF3/ZR

3 is finite in the perturbation expansion. The value of ZR can bedetermined if we know the on-shell residue of the two-point renormalized Green’sfunction

G(2)R (k) ≈

k2→m2F

iZR

k2 − m2F + iε

(4.24)

or, equivalently∂

∂k2�

(2)R (k)|k2=m2

F= 1

ZR(4.25)

Due to the relation

G(n)F (x1, . . . , xn) = Z−n/2

R G(n)R (x1, . . . , xn) (4.26)

the factors of ZR will also appear in the relation between the renormalized Green’sfunctions and the S-matrix elements. This does not make the renormalized Green’sfunctions very ‘unphysical’ because the ZR factors are finite order-by-order.

The relation between the S-matrix and the renormalized Green’s functions hasthe following form

〈k′1 . . .k′m |S|k1 . . .kn〉 = limk2

i →m2F

∏i

[1

i(k2

i − m2F)

]×Z−(n+m)/2

R G(n+m)R (k ′1, . . . , k ′m,−k1, . . . ,−kn) (4.27)

which generalizes (2.190).Of course, the lagrangian (4.15) can be rewritten in terms of the new renormal-

ized fields and parameters

L = L(Z3�R, (Z0/Z3)m2R, (Z1/Z2

3)λR, . . .)

= L(�R,m2R, λR, . . .)+ L (4.28)

where in the λ�4 theory

L = 12(Z3 − 1)(∂µ�R)

2 − 12(Z0 − 1)m2

R�2R − 1/4!(Z1 − 1)λR�

4R (4.29)

The Feynman rules of the renormalized perturbation theory follow from therenormalized lagrangian (4.28). The counterterms provide vertices, with Z -dependent coupling constants: (Z3 − 1)p2, (Z0 − 1)m2

R, (Z1 − 1)λR. Note thatthe counterterms are treated strictly perturbatively: when calculating the nth ordercorrection it is enough to know the counterterms up to this order only.

The counterterms are constructed according to a renormalization prescription.One possibility is the ‘physical’ renormalization prescription described before. A


more general case corresponds to specifying finite values for some renormalizedGreen’s functions at a renormalization point. For instance (�R is defined by (4.22))

∂

∂k2�

(2)R (k)|µ = 1, �R(k)|µ = 0, �

(4)R (k)|µ = λR (4.30)

Clearly the number of the equations must be equal to the number of the renormal-ization constants.

Another possibility is to assume some trial form for the counterterms and adjustthe parameters to make the renormalized Green’s functions finite order-by-order.This is the case in the minimal subtraction scheme. This procedure is particularlyconvenient because the counterterms can be determined from the divergent parts ofthe diagrams alone. We shall discuss this scheme in detail in Section 4.3, after theintroduction of dimensional regularization.

Final remarks

Let us finish this section with some remarks concerning the calculation of thescattering amplitudes in the renormalized perturbation expansion. As explainedbefore, we determine the physical mass m2

F from the requirement that the inversepropagator �(2)

R (k) = k2−m2R−�R(k) vanishes at k2 = m2

F. We use this conditionin preference to the one involving the propagator G(2)

R (k). The reason is that theperturbation expansion for G(2)

R (k) has the form

G(2)R (k) = i

k2 − m2R

+ i

k2 − m2R

[−i�R(k)]i

k2 − m2R

+ i

k2 − m2R

[−i�R(k)]i

k2 − m2R

× [−i�R(k)]i

k2 − m2R

+ · · · = i

k2 − m2R −�R(k)

(4.31)

corresponding to the diagrams

with the blobs on the r.h.s. denoting −i�R(k). We see that, to any finite order,G(2)(k) has multiple poles at k2 = m2

R instead of a single pole at k2 = m2F required

by unitarity. To obtain a single pole one must sum the whole geometric series(4.31).

Assume that �(2)R (k) calculated up to order n in some renormalization scheme is

regular at k2 = m2F and expand

�R(k2) = �0 + (k2 − m2

F)�1 + (k2 − m2F)

2�2 + · · · (4.32)

where m2F is understood as the value of k2 at which the nth order approximation to

4.2 Classification of the divergent diagrams 157

�(2)R (k) vanishes; hence �0 = m2

F − m2R. We obtain

�(2)R (k) = (k2 − m2

F)[1 −�1 − (k2 − m2F)�2 − · · ·] (4.33)

and, using (4.25)

ZR = 1

1 −�1(4.34)

Consequently, (4.27) becomes

〈k′1 . . .k′m |S|k1 . . .kn〉= lim

k2i →m2

F

∏i

(1

i(k2

i − m2F)

)(1 −�1)

(n+m)/2

×G(n+m)R (k ′1, . . . , k ′m,−k1, . . . ,−kn)

= (1 −�1)−(n+m)/2G(n+m)

R(trunc) (4.35)

In the lowest non-trivial order (one-loop approximation) we can equivalentlyreplace (1−�1)

−(n+m)/2 by (1+ 12(n +m)�1) so that we then have the correction

(+ 12�1) for each external line. We shall often refer to this rule in one-loop

calculations.

4.2 Classification of the divergent diagrams

Structure of the UV divergences by momentum power counting

In the previous section we have described the procedure of renormalization.However, we have not given the proof that this procedure actually leads to finiteresults: we still have to show that a theory like λ�4 is renormalizable. In otherwords, we should still prove that the counterterms induced by the renormalization,which are of the same general structure as vertices of the original lagrangian, areenough to cancel UV divergences due to loop momentum integrations. Here weshall give only a general idea of the proof. We shall concentrate on the structureof UV divergences which appear in Feynman diagrams of λ�4 perturbation theory(with counterterms not yet included) and use the results to determine the formof counterterms from the requirement that these divergences are cancelled. Theyappear to have the same form as the counterterms induced by the renormalizationdescribed in Section 4.1. Some examples from QED and λ�3 theories will also beconsidered.

The basic blocks of a renormalization programme are the connected propervertex functions �(n), also called the 1PI Green’s functions. Let us recall thatthese are truncated Green’s functions which remain connected when an arbitraryinternal line is cut. In a renormalization programme these are important for two


reasons. First, any other Green’s function is made up of 1PI Green’s functionsjoined together to form a tree, each loop included entirely in one of the 1PI parts,so that the momentum integrations in the two distinct parts are independent of eachother. Consequently, all loops and therefore all divergences are present alreadyin the 1PI Green’s functions. The second reason is that the identification of thenecessary counterterms which are to be interpreted as additional vertices in therenormalized lagrangian is particularly simple if one looks at the divergences ofthe 1PI functions. This is because the connected proper vertex functions canthemselves be understood as generalized vertices of an effective lagrangian whichincludes all loop corrections: the generating functional �[�].

The UV divergences we are concerned with come from the region of integrationwhere some of the integration momenta are large. Some information about thepossible structure of these divergences can be obtained by the simple procedureof counting the powers of momentum in the Feynman integrand. Suppose that allintegration momenta become large simultaneously: ki → �ki ,� → ∞. Theintegrand will then behave as a power of �, say �−N . One defines the so-calledsuperficial degree of divergence of a Feynman diagram

D = 4L − N (4.36)

where L is the number of independent loops, each corresponding to a∫

d4ki

momentum integration. For a diagram with V vertices and I internal lines thenumber of independent loops is L = I − V + 1.

Let us next assume that only some subset S of the integration momenta is largeand of the order �. The integrand will now be proportional to another powerof �:�−NS . We can introduce the superficial degree of divergence DS of thissubintegration: DS = 4L S − NS , where L S is the number of integration momentabelonging to S. It is easy to see that DS can also be regarded as a superficialdegree of divergence of a subdiagram built up of all the internal lines of thecomplete Feynman diagram with momenta depending on those in S: L S is thenthe number of loops in this subdiagram. In this way we can assign a subdiagramand a degree of divergence to each loop subintegration. For example, in the case ofthe two-loop diagram (a) in Fig. 4.2 the subdiagrams (b) and (c) correspond to thesubintegrations over l and k, respectively.

A theorem due to Weinberg, states that a Feynman integral converges if thedegree of divergence of the diagram as well as the degree of divergence associatedwith each possible subintegration over loop momenta is negative. This fundamentalresult will be the basis of our discussion. The first thing to note is that in λ�4 theory(and any other interacting quantum field theory in four dimensions) diagrams withD ≥ 0 are present. In the above example diagram (a) has D = 2 while (b) and (c)both have D = 0.


Fig. 4.2.

Classification of divergent diagrams

To find out what counterterms are required we must examine all 1PI Green’sfunctions with D ≥ 0 which appear in the theory. Consider the case of a scalarfield theory without derivative terms in the interaction (λ�4 is one example). Then,for any 1PI diagram (4.36) reads

D = 4L − N = 4L − 2I (4.37)

where I is the number of internal lines. In this formula the restriction to 1PIdiagrams is due to the fact that propagators corresponding to the internal lineswhich do not belong to loops are independent of the integration momenta andconsequently do not contribute to N : in 1PI diagrams such lines are absent. Wecan also write a generalization of (4.37) which is applicable to the case of fermionQED, again with restriction to 1PI diagrams

D = 4L − 2IB − IF (4.38)

Here IB, IF are the numbers of photon and fermion internal lines, respectively. Notethat by replacing 4L with nL we obtain a generalization of (4.36), (4.37) and (4.38)to the case of an n-dimensional space.

We shall see that D is determined by the number of external lines of the diagram.The argument is based on dimensional analysis so let us first recall the dimensionsof some quantities in quantum field theory. We set h = c = 1 so that theaction integral must be dimensionless, while length and mass are measured ininverse units: [x]−1 = [m]. The canonical dimensions of Bose and Fermi fieldsare determined from the requirement that the kinetic terms in the action, whichare, respectively, of the form

∫dnx �∂2� and

∫dnx �i/∂�, are dimensionless. It

follows that

[�] = [m](n−2)/2, [�] = [m](n−1)/2 (4.39)

In the same manner the dimensions of a coupling constant follow from therequirement that the corresponding term in the action is dimensionless. Some


examples are

λ�3: [m]−n[λ][m]3(n−2)/2 = 1; [λ] = [m]3−n/2

λ�4: [m]−n[λ][m]4(n−2)/2 = 1; [λ] = [m]4−n

g�A/�: [m]−n[g][m]2(n−1)/2+(n−2)/2 = 1; [g] = [m]2−n/2

(4.40)

where [m]−n is the dimension of the∫

dnx integration measure. In four dimensionsthe quartic (λ�4) and QED (g�A/�) coupling constants are dimensionless whilethe cubic (λ�3) coupling constant has the dimension of mass. Assume that atlarge momenta the Bose and Fermi propagators behave, respectively, as 1/k2 and1/k. Then, for any 1PI Feynman diagram built up of vertices with dimensionlesscoupling constants only, the superficial degree of divergence coincides with thedimension of the diagram in the unit of mass. The exception is the possiblelowest order no-loop contribution, i.e. the single elementary vertex for whichthe superficial degree of divergence is undefined. As all contributions to agiven Green’s function must be of the same dimension it then follows that withdimensionless couplings, the superficial degree of divergence of a diagram isdetermined solely by the external lines. The canonical dimension of a 1PI diagramwith B external boson lines and F external fermion lines is equal to the dimensionof a coupling constant corresponding to the vertex of this structure, which is[m]4−B−3F/2 in four dimensions. Consequently

D = 4 − B − 32 F (4.41)

As explained, this applies to 1PI diagrams with dimensionless couplings only. Ifthe diagram also contains vertices with dimensional coupling constants of totaldimension in the units of mass (for example, = K for K fermion massinsertions or cubic scalar couplings)

D = 4 − B − 32 F − (4.42)

(also see Problem 4.4). This formula has some important consequences. One isthat in any theory with coupling constants all of positive dimension in the units ofmass (λ�3 for example) the number of 1PI diagrams with D ≥ 0 is finite. A theorylike this is called super-renormalizable. In the case of the λ�3 theory the diagramswith D ≥ 0 are as shown in Fig. 4.3. Here (a), (b) and (c) are irrelevant vacuumdiagrams: they cancel with the normalization factor of our Green’s functions, (d)is a constant ‘tadpole’ which contributes to the vacuum expectation value 〈�〉of the � field and can be disregarded in the 〈�〉 = 0 case (diagrams of thiskind will be important in theories with spontaneous symmetry breaking) and (e)produces an infinite mass renormalization counterterm which is the only divergentcounterterm in λ�3 theory with 〈�〉 = 0. In particular, there are no D ≥ 0 1PI


Fig. 4.3.

diagrams with three external lines and consequently no divergent coupling constantrenormalization counterterm.

On the other hand in a theory which involves, among others, any couplingconstant of negative dimension, like λ�4 in more than four space-time dimensionsor G(��)2 in four-dimensional space, the perturbation expansion of any 1PIGreen’s function must include D ≥ 0 diagrams when the order is high enough.Each 1PI Green’s function requires new counterterms and new renormalizationconditions: consequently, the theory is non-renormalizable. An important exampleis Fermi’s theory of weak interactions.

Let us now return to theories with dimensionless coupling constants, so that(4.41) applies: all diagrams contributing to a specific 1PI Green’s function havethe same degree of divergence. For the λ�4 theory and for QED we have theD ≥ 0 connected proper vertex functions shown in Fig. 4.4 (the triple photonproper vertex also has D = 1 but it vanishes because the photon has odd chargeconjugation).

Necessary counterterms

To make the divergent diagrams finite we must find a way to lower the associateddegrees of divergence. This can be achieved by making a subtraction in theFeynman integrand. Consider a 1PI Feynman diagram with D ≥ 0. Let { f }D

denote the sum of first (D + 1) terms of a Taylor expansion of the Feynmanintegrand f in powers of the external momenta with D equal to the superficialdegree of divergence of the diagram considered. If we replace the integrand of thisdiagram by a subtracted expression

f − { f }D (4.43)


Fig. 4.4.

the superficial degree of divergence will become negative. A similar procedureshould be applied to divergent subdiagrams.

In a renormalizable theory we should be able to understand these subtractions asdue to the inclusion of counterterms. With some procedure for the regularizationof divergent integrals we can calculate the Feynman integrals of the two terms in(4.43) separately. As the second term is a polynomial in the external momenta,after the integration it will produce a polynomial ‘counterterm’, or a combinationof counterterms. Obviously the counterterms will diverge when the intermediateregularization is removed; however, from Weinberg’s theorem, they must combinewith the other part of (4.43) to ultimately give a finite result provided that we havedealt with all divergent subdiagrams.

Consider the case of the λ�4 scalar field theory. We then have two D ≥ 0Green’s functions: �(2)(p) with D = 2 and �(4)(p1, . . . , p4) with D = 0. The two-point function has D = 2 and consequently requires subtraction of the first threeterms of the Taylor expansion of the integrand. This produces two counterterms

Ap2 + B

The third one, proportional to pµ, vanishes because of relativistic invariance: allcounterterms must be scalars. We recognize the counterparts of the wave-functionrenormalization constant (p2 is equivalent to −∂2) and the mass counterterm. Thefour-point function has D = 0 and only one (constant) counterterm which corre-sponds to the coupling constant counterterm. We conclude that the subtractionsnecessary to lower the degree of divergence of the D ≥ 0 Green’s functions are


Fig. 4.5.

Fig. 4.6.

equivalent to including the counterterms of structure as implied by renormalizationof the original lagrangian.

Can these counterterms, if included in the interaction lagrangian, also deal withall divergences due to the D ≥ 0 subdiagrams? Again (in the λ�4 case) theseare the subdiagrams with two (D = 2) or four (D = 0) external lines, so that nonew types of counterterm are apparently required. We can envisage the followingprocedure. Consider a diagram with no counterterm vertices which includes someD ≥ 0 subdiagrams. We first identify all lowest order (one-loop) divergentsubdiagrams. In each of these we make the appropriate subtractions. The effectof these subtractions should be equivalent to adding to the original diagram alldiagrams obtained by replacing some of the considered lowest order subdiagramsby the corresponding lowest order counterterms. One example is shown in Fig. 4.5.The boxes indicate the divergent subdiagrams we are considering. Note that in thiscase the dot is not the complete lowest order coupling constant counterterm.This is because we make the subtraction in the indicated subdiagrams, and not incomplete one-loop vertex functions which are shown in Fig. 4.6. This procedureshould then be repeated with respect to higher order D ≥ 0 subdiagrams andcounterterms until all divergences are removed.

Unfortunately, this argument runs into difficulties: one problem is that we cannotalways make subtractions independently in different subdiagrams. For instance,subdiagrams (b) and (c) in Fig. 4.2 overlap each other. Another example ofsuch ‘overlapping divergences’ is the QED diagram shown in Fig. 4.7, where


Fig. 4.7.

the D = 0 divergent subdiagram can be identified in two distinct ways. Theproblem of overlapping divergences requires more detailed treatment and we shallnot discuss it here. However, the ultimate answer is positive: the counterterms inthe renormalized lagrangian do indeed cancel all UV divergences.

We must also note another approach, the BPHZ construction (Bogoliubov–Parasiuk–Hepp and Zimmermann). In this case the subtractions are performedsystematically by means of an ‘R-operation’ which is based on a somewhatdifferent identification of subdiagrams.

4.3 λ�4: low order renormalization

Feynman rules including counterterms

In this section we introduce in some detail the dimensional regularization method(’t Hooft & Veltman 1972, Bollini & Giambiagi 1972) and the minimal subtractionrenormalization scheme using the λ�4 theory as an example. We choose theminimal subtraction scheme to illustrate our general consideration of Section 4.1since this method of renormalization is particularly convenient in applicationsof the renormalization group equation: this is the so-called mass-independentrenormalization, namely the renormalization constants Zi do not depend on massparameters.

The Feynman rules for the λ�4 theory are collected in Table 4.1. It also showsdiagrams corresponding to the counterterms present in lagrangian (4.28). Allcounterterms are treated as interaction terms since we want to use countertermsorder-by-order in the perturbation theory.

Renormalization constants Zi will be determined by the renormalization condi-tions in each order of the perturbation theory

Zi = 1 +∑

n

(Zi − 1)(n)

4.3 λ�4: low order renormalization 165

Table 4.1.

propagatori

p2 − m2 + iε

loop integration∫

d4k

(2π)4

vertex −iλ; p + q + r + s = 0

symmetry factors S S = 12

S = 16

S = 12

S = 14

vertex counterterm −iλ(Z1 − 1)(n)

mass counterterm −i(Z0 − 1)(n)m2

wave-functionrenormalization +i(Z3 − 1)(n) p2

counterterm

where

(Zi − 1)(n) = f (n)i λn

It is clear from the previous section that the main task for the lowest orderrenormalization is to calculate the proper (1PI) Green’s functions correspondingto the three divergent diagrams of Fig. 4.8. The calculation will be based on thedimensional regularization method and we introduce it now briefly. We begin withvertex renormalization, i.e. with Fig. 4.8(b). In four space-time dimensions this


Fig. 4.8.

Fig. 4.9.

is logarithmically divergent ∼ ∫d4k/k4. However, if we do the integration in

4 − ε dimensions the result is finite. This suggests the possibility of consideringthe analytic continuation of the amplitude to 4 − ε dimensions (regularization),cancelling all terms divergent when ε → 0 by properly adjusting the renormaliza-tion constants Zi (renormalization) and continuing the result analytically to fourdimensions. Thus, the dimensionally regularized theory (4.1) is defined by theaction

S =∫

dnx

[12(∂µφ)

2 − 12 m2φ2 − λµε

4!φ4 + counterterms

]where n = 4−ε. In n = 4−ε dimensions the coupling constant λ is dimensionful,so it is convenient to write it as µελ, where µ is an arbitrary mass scale and λ isdimensionless. The Feynman rules in Table 4.1 are changed accordingly.

Calculation of Fig. 4.8(b)

We first calculate the diagram shown in Fig. 4.8(b) in 4 − ε dimensions and thendiscuss some subtle points of the method. The kinematics are defined in Fig. 4.9.The first diagram of Fig. 4.9 gives the following expression for the amplitude:

A1 = 1

2

∫dnk

(2π)n

i

k2 − m2 + iε

i

(k − P)2 − m2 + iε(−iλµε)2 (4.44)


Fig. 4.10.

We introduce the standard Feynman parameters

1

ab=∫ 1

0

dx

[ax + b(1 − x)]2(4.45)

and change the variables as follows

k − Px → k (4.46)

which is legitimate for a convergent integral. We then get

A1 = 12λ

2µ2ε∫ 1

0dx∫

dnk

(2π)n

1

[k2 + sx(1 − x)− m2 + iε]2(4.47)

The integration over dk0 dn−1k can be easily done after the Wick rotation. We notethat the poles of the integrand in the k0-plane are located as shown in Fig. 4.10,since k0 = ±[k2− sx(1− x)+m2− iε]1/2. We can change the contour to integrateover the imaginary axis. Changing variables

ik0 → k0

dk0 →−i dk0

k20 − k2 →−k2

0 − k2∫ −i∞

+i∞dk0 →+i

∫ ∞

−∞dk0

one gets the following expression

A1 = 12λ

2µ2ε∫ 1

0dx∫

dnk

(2π)n

(+i)

[k2 − sx(1 − x)+ m2 − iε]2(4.48)


where k is a vector in n-dimensional Euclidean space. The next step is to use theformula (in Euclidean space)∫

dnk

(2π)n

1

(k2 + b)α= 1

(4π)n/2

bn/2−α�(α − 12 n)

�(α)(4.49)

which leads us to an almost final result

A1 = 12(λµ

ε)iλ

(4π)n/2

�( 12ε)

�(2)

∫ 1

0dx∫ [

m2 − sx(1 − x)− iε

µ2

]−ε/2

(4.50)

To proceed we require the expression

�(x)−→x→0

1/x − γ + O(x)

where γ = limn→∞(1+ 12 + · · ·+ 1/n− ln n) = 0.577 is Euler’s constant, and we

also use the expansion

aε = exp(ε ln a) = 1 + ε ln a + O(ε2)

In the limit ε → 0 we get

A1 = iλµε λ

16π2

1

ε− iλµε λ

2(4π)2

[ ∫ 1

0dx ln

m2 − sx(1 − x)

µ2+γ − ln 4π

]+O(ε)

(4.51)The presence of the pole term 1/ε reflects the divergence of the integral in fourdimensions. Note also the presence in (4.50) and (4.51) of the factor λµε (notexpanded in ε) since this is the coupling constant in n dimensions and it enters intothe definition of the counterterm −iλµε(Z1 − 1). The remaining contribution isfinite and reads, after integration,

A1(s) = −i(λµε)λ

2(4π)2

{γ − ln 4π − 2 + ln

m2

µ2

+ 2(

14 − m2/s

)1/2ln

[12 +

(14 − m2/s

)1/2

12 −

(14 − m2/s

)1/2

]}(4.52)

The inclusion of the crossed diagrams gives the final result

A1 = i(λµε)λ3

16π2ε

1

ε+ A1(s)+ A1(t)+ A1(u) (4.53)

Comments on analytic continuation to n �= 4 dimensions

Before proceeding further a comment on the explored analytic continuation ton �= 4 dimensions should be made. First, consider the space with one time-like


dimension and n − 1 �= 3 space-like dimensions. The basic integral

In =∫

dnk1

(k2 + b)2

makes sense for n = 1, 2, 3 and can be written as

In = 1

2

∫ ∞

−∞dk0

∫ ∞

0dr2 (r2)(n−3)/2 1

(k2 + b)2

∫dn−1�, k2 = k2

0 − r2 (4.54)

where dn−1� is an element of the solid angle in (n − 1) dimensions

�n−1 =∫ 2π

0dθ1

∫ π

0sin θ2 dθ2

∫ π

0sin2 θ3 dθ3 . . .

∫ π

0sinn−3 θn−2 dθn−2

= 2π(n−1)/2

�(

12(n − 1)

)Integral (4.54) can be used to define an analytic function In in the range 1 < n < 4which coincides with the original In for n = 2, 3 (the solid angle for a non-integern is the analytic continuation of �n−1, n integer). Furthermore, by partiallyintegrating N times over dr2 one can find explicit expressions for the analyticcontinuation of In to (1 − 2N ) < n < 4 dimensions. In addition we would like tobe able to continue In to values n > 4. This can be achieved using the identity

I = 1

2

(d

dk0k0 + d

drr

)and by integrating each term by parts over k0 and r , respectively. For 1 < n < 4the surface terms are zero and one gets

In ∼∫ ∞

−∞dk0

∫ ∞

0dr rn−2 (−2k2

0 + 2r2)

(k2 + b)3

+ 1

2

∫ ∞

−∞dk0

∫ ∞

0dr rn−2 (n − 2)

(k2 + b)2= I ′n + 1

2(n − 2)In (4.55)

so that

In = − 1

n − 4I ′n

where I ′n exists for 1 < n < 5. The above procedure may be repeated to show thatIn is of the form

In = �(

12(4 − n)

)In (4.56)

where In is well behaved for arbitrarily large n. So one has constructed an analyticfunction In with simple poles at n = 4, 6, 8, . . . . Dimensional regularizationassumes the physical theory to be given by In for n → 4.


Fig. 4.11.

Lowest order renormalization

Coming back to the result (4.53) one can now renormalize the amplitude byincluding counterterms. It is clear from the structure of the counterterm that thedivergences of (4.53) can be cancelled by the vertex counterterm in Fig. 4.11, i.e.by properly adjusting Z1. The minimal subtraction (MS) prescription assumes thatZi have only (multiple) pole terms in ε and the residua are fixed to cancel all thedivergences for ε → 0. With such a prescription one then gets to order λ

(Z1 − 1)(1) = 1

ε

3λ

16π2(4.57)

and the renormalized (finite) amplitude of Fig. 4.8(b) is given by A1(s)+ A1(t)+A1(u) of (4.53). The so-called MS scheme in which

(Z1 − 1)(1) = 1

2

3λ

16π2

(2

ε− γ + ln 4π

)is also often used.

The other two diagrams of Fig. 4.8 require mass and wave-function renormal-ization. The amplitude of Fig. 4.8(a) reads

12(−iλ)µε

∫dnk

(2π)n

i

k2 − m2 + iε

= 12(−iλ)µε 1

(4π)2(4π)ε/2(m2)1−ε/2�

(12ε − 1

)= 1

2 i2λm2

(4π)2

1

ε− 1

2iλm2

(4π)2

(−1 + γ − ln 4π + ln

m2

µ2

)(4.58)

The divergence is cancelled by the mass counterterm with

≡ (−i)(Z0 − 1)(1)m2

(Z0 − 1)(1)λ

16π2= 1

ε(4.59)

in the minimal subtraction renormalization scheme.The amplitude of Fig. 4.8(c) requires both mass and wave-function renormal-

ization. It is of order λ2 and therefore in order-by-order renormalization onemust include simultaneously the other diagrams of order λ2 given in Fig. 4.12.


Fig. 4.12.

The divergences which remain after adding the amplitude of Fig. 4.8(c) and thoseof Fig. 4.12(a)–(c) should be then cancelled by new λ2 terms of the mass andwave-function renormalization counterterms

≡ −i(Z0 − 1)(2)m2

≡ +i(Z3 − 1)(2) p2

(notice that in λ�4 theory (Z3 − 1)(1) = 0).We tabulate first the results for Fig. 4.12(a)–(c). Using (4.57) and (4.59) it is

easy to get the following (we write down explicitly only pole terms):

Diagram 4.12(a):

iλ2 m2

(16π2)2

3

2

(2

ε2− 1

εln

m2

4πµ2+ 1

ε− γ

ε

)+ O(ε0) (4.60a)

Diagram 4.12(b):

iλ2 m2

(16π2)2

1

2

(2

ε2− 1

εln

m2

4πµ2− γ

ε

)+ O(ε0) (4.60b)

Diagram 4.12(c):

(−i)λ2 m2

(16π2)2

1

2

(2

ε2− 2

εln

m2

4πµ2+ 1

ε− 2γ

ε

)+ O(ε0) (4.60c)

The only non-trivial contribution is due to Fig. 4.8(c). The amplitude reads

I = 16λ

2µ2εi∫

dnk

(2π)n

dnl

(2π)n

1

k2 − m2 + iε

1

l2 − m2 + iε

1

(p + l − k)2 − m2 + iε(4.61)

which is divergent for (i) l fixed, k large, (ii) k fixed, l large, (iii) both k and l large.This is what is called an overlapping divergence because several overlapping loopsubintegrations lead to a divergent result. The calculation of amplitude (4.61) islengthy (Collins 1974). The structure of pole terms turns out to be the following

I = iλ2

(16π2)2

{−m2

ε2+ 1

ε2

[m2 ln

m2

4πµ2+ 1

12p2 + (γ − 3

2)m2

]}


(1) (1)(1)+ + +

(1)

Fig. 4.13.

Adding (4.60) one gets

iλ2

(16π2)2

(1

ε22m2 + 1

ε

1

12p2 − 1

ε

1

2m2

)so that

(Z0 − 1)(2) = λ2

(16π2)2

(2

ε2− 1

2

1

ε

)(Z3 − 1)(2) = − λ2

12(16π2)2

1

ε

(4.62)

Combining (4.59) with (4.62) one gets the following results for Z0 and Z3 (to orderλ2)

Z0 = 1 + λ

16π2

1

ε+ λ2

(16π2)2

(2

ε2− 1

2

1

ε

)(4.63)

Z3 = 1 − λ2

12(16π2)2

1

ε(4.64)

In the literature one can also often find the constants Zm and Zλ defined by therelations

m2B = Zmm2

R, i.e. Zm = Z0/Z3

λB = ZλλR, i.e. Zλ = Z1/Z23

}(4.65)

We see in particular that to order λ2

Zm = 1 + 1

ε

[λ

16π2− 5

12

λ2

(16π2)2

]+ 1

ε2

2λ2

(16π2)2(4.66)

Notice that there are logarithms of mass in the residua of the poles in each diagram.However, they cancel in the final result for constants Zi and we see explicitly that atleast in that order of perturbation theory the minimal subtraction scheme is indeeda mass-independent scheme.

We end this section with two comments. Firstly, it should be remembered thatone-particle reducible diagrams do not require new counterterms. Let us illustratethat fact with one example. Consider the diagrams of Fig. 4.13. We observe that

4.4 Effective field theories 173

the following relation holds.

Diagrams of Fig. 4.13 =(

+ (1))2

which illustrates our statement.The second remark is devoted to the fact that the finite parts of Feynman integrals

are not, in general, obtained by simply discarding poles and multiple poles at n = 4but have to be worked out by order-by-order renormalization procedure.

Consider, for example, mass loop diagrams

≡ (1/ε) f1 + f2 + ε f3

(1) ≡ −(1/ε) f1

so that in the second order(+ (1)

)2

≡ f 22

However, for the two-loop diagram alone we have

≡ (1/ε2) f 21 + f 2

2 + ε2 f 23 + 2(1/ε) f1 f2 + 2 f1 f3 + 2 f2 f3ε

Discarding poles in the last result one obtains

f 22 + 2 f1 f3 �= f 2

2

which is wrong because we insist on unitarity being satisfied order-by-order inperturbation theory and therefore the finite part of the two-loop diagram shouldread f 2

2 as can be seen by considering the imaginary part of the amplitude, see(3.70).

4.4 Effective field theories

The basic tool for calculations in quantum field theory is perturbation theory. Inperturbative calculations we face the problem of divergent results, typical for localfield theories. In fact, this problem is also present in classical field theories. Thepredictive power of a theory is retained due to the procedure of renormalization:divergent contributions are absorbed into free parameters of the theory, which aretaken from experiment. Thus, we are still able to predict results of measurementsin terms of other measurable quantities.

The number of free parameters is related to the number of so-called countertermsin the lagrangian, which are required for the procedure of renormalization, i.e. toabsorb all infinities. It is very important to realize that this number may be finite


or infinite! Traditionally, a theory is called renormalizable only if a finite numberof counterterms is sufficient to renormalize it. This certainly makes such a theoryvery predictive: all results are obtained in terms of a finite (usually small) numberof measurable parameters. For a theory to be renormalizable in this traditionalsense, the corresponding lagrangian may contain only operators of dimension four,or less, i.e. it cannot contain couplings with the dimension of negative powers ofmass. We know that such theories are often very good approximations to the realworld in some energy range and therefore very interesting to study. However, wealso know now that they should be regarded as only effective theories, valid in alimited energy range.

Effective field theories appear naturally in the physical situation characterizedby the presence of two hierarchical mass scales M � m. Suppose that physicsat scale M � m is described by a renormalizable theory. At the energy scalem, after integrating out heavy degrees of freedom (see Chapters 7 and 12), oneobtains an effective theory whose lagrangian may consist of operators of dimensionfour or less (the ‘renormalizable’ part) and higher dimension operators suppressedby inverse powers of large mass M . If we neglect all physical effects of thosecorrections, then physics at m is described by the ‘renormalizable’ part. A well-known example is, for instance, quantum electrodynamics (QED) considered as theeffective low energy part of the electroweak theory (Chapter 12). It is the contentof the Appelquist–Carrazone decoupling theorem (Appelquist & Carrazone 1976)that all effects depending on positive powers of M or logarithmic in M can beabsorbed into the redefinition of the bare parameters of the effective renormalizabletheory.

As we shall see, we may, however, be interested in the O(m/M) corrections,too. We encounter the latter case, for example, if we want to calculate QED andQCD corrections to low energy, E ∼ 1 GeV, weak processes (Chapter 12). Theseare described by the weak effective lagrangian with dimensionful coupling. SinceE/MW � 1, it is sufficient to use it only in the Born approximation, with no needfor higher order perturbation in it. In this case, the virtue of the effective theory isnot only that it offers a quickly truncated expansion in powers of E/MW but, firstof all, that it provides a tool for easy resummation of large logarithmic correctionsO(αn lnn(E/MW)) (Chapter 12) to weak processes.

We have described here a situation in which the complete renormalizable theoryat the scale M is known and the effective theory can be derived from it byperturbative methods. A different situation occurs when we try to describe stronginteractions of hadrons at low momentum transfer. Perturbative QCD is notapplicable in that region but we may construct, for instance, an effective pionnucleon theory which respects the chiral symmetry of QCD but is not derivableperturbatively from it. Moreover, the physical cut-off � for that theory is of similar

Problems 175

order of magnitude as the maximal energy of its applicability so that the E/�

expansion cannot be so quickly truncated and higher order iteration of the effectivelagrangian is necessary. Therefore, it must be stressed that there is no fundamentaldifference between a theory which is renormalizable with a finite or infinite numberof counterterms (but only those consistent with the symmetry of the bare action).The latter still has a lot of predictive power (based on such principles as analyticity,unitarity, symmetries), although the number of free parameters increases with theorder of perturbation theory.

Finally, we may consider the physical situation when we know about (or expect)the existence of a large physical scale (for example, Planck scale) but we do notknow the correct theory at that scale. It is then a very natural hypothesis thatphysics at lower scales is described by an effective theory which contains allhigher dimension operators consistent with symmetries of the low energy theoryand suppressed by powers of the large scale.

It is of fundamental importance that Nature can be described by field theoriesthat obey the decoupling theorem. This makes it possible to understand fundamen-tal laws of physics step by step, without the (unrealistic) quest for the theory ofeverything. On the other hand, it makes it difficult to learn about new physics athigher scales, as such effects are always strongly suppressed.

It should be emphasized that, although a renormalizable theory can formally beused for calculations at any energy scale (since the dependence on larger scales istotally absent), this does not mean that it is correct at any scale. A good exampleis again QED. At the scale MZ it must be embedded into the electroweak theory.However, we cannot learn about that by studying electromagnetic processes at lowenergy, E � MZ , unless we have reached the precision of sensitivity to O(E/MZ )

corrections.The above comments on effective field theories anticipate a lot of knowledge

which the reader is going to acquire in the following chapters. However, they maybe a useful guide to further reading.

Problems

4.1 Discuss a model with two scalar fields φ and �, with the masses m and M ,respectively, such that m � M .

(a) Construct the most general renormalizable lagrangian invariant under the symme-try φ →−φ, φ →−�.

(b) Study spontaneous breaking of this discrete symmetry at the classical level.(c) Assuming that the discrete symmetry is spontaneously broken calculate the

vacuum expectation value v at one-loop level; calculate the one-loop effectivepotential by integrating the tadpole amplitudes. Hint: expand φ = φ − v where


〈φ〉 = v is to be found from the minimum of the effective potential; rewrite thelagrangian in terms of the φ; the presence of the φ3 coupling generates tree level( ) and one-loop ( ) tadpole diagrams; the vacuum expectationvalue v is fixed by the equation ( + = 0); calculate theone-loop contribution V1 to the effective potential by integrating the equation V1/dv = i and identifying φ ≡ v.

(d) Single out from the effective potential the corrections to the mass m and to theself-coupling of the field φ which depend on the heavy field � and compare themwith the result of the direct diagrammatical calculation.

(e) Formulate the effective theory at the scale µ � M ; pay attention to the structureof quantum corrections from the exchange of the field �.

4.2 Discuss a model with a scalar field φ and a fermion field ψ , with the masses M andm, respectively, such that m < M .

(a) Construct the most general renormalizable lagrangian.(b) Calculate one-loop self-energy diagrams for the field ψ and compare the cor-

rections to its mass with the analogous corrections to the mass of the field inProblem 4.1.

4.3 It is convenient to perform multi-loop calculations in Euclidean space. Instead ofperforming Wick’s rotation in the Feynman integrals it is more convenient to beginwith the theory formulated in the Euclidean space. For the �4 theory we then have thegenerating functional

W [J ] =∫

[d�] exp

[−SE −

∫d4

E x J (x)�(x)

](4.67)

where

SE =∫

d4E x

[1

2(∂�)2 + 1

2m2�2 + λ

24�4]

(4.68)

Check that the n-point Green’s functions in the Minkowski space are obtained fromthose in the Euclidean space by analytic continuation of their momentum variables:p4

i → −ip0i , where p4 is the ‘time’ component of the momentum four-vector in the

Euclidean space.

4.4 Calculate the superficial degree of divergence in n dimensions of a Feynman diagramwith I f (E f ) internal (external) lines of type f and Ni vertices of type i , with di

derivatives. Show that D =∑f I f (2s f − 2+ n)+∑

i Ni (di − n)+ n, where s f = 0(1/2) for spin 0 (1/2) particles. Using 2IF+E f =

∑i ni f Ni , where ni f is the number

of lines of type f in the vertex of type i , one gets D = n −∑f E f (s f − 1 + n/2)−∑

i Ni i with i = n−di −∑

f ni f (s f −1−n/2). Only the last term in the equationfor D depends on the internal structure of diagrams. If all i ≥ 0 then D ≤ Dmax andthe theory is renormalizable with a finite number of counterterms.

5

Quantum electrodynamics

We begin our discussion of gauge theories with the simplest and the best-knownone: QED. So far most of the physical applications of gauge theories rely onperturbation theory, crucial for which is the renormalization programme describedin Chapter 4 which is usually formulated in terms of the Green’s functions.As we know from Chapter 3, to derive the Feynman rules for a gauge theoryit is necessary to fix the gauge by adding some gauge-fixing conditions. Allphysical quantities, and the S-matrix elements in particular, are gauge-independent.This fact, which is necessary for a consistent theory, has to be checked in eachcase. The Green’s functions are not, however, physical objects and they aregauge-dependent. Nevertheless, the gauge symmetry of the theory leads to certainconstraints on the Green’s functions: there exist relations between them known asthe Ward–Takahashi identities. In terms of the Green’s functions these are exactlythe conditions which protect the physical equivalence of different gauges.

The QED lagrangian is based on the minimal coupling ansatz and in terms ofthe bare fields and parameters it reads

L = − 14 FB

µν + Fµν

B + �B(i/∂ − qeB A/B)�B − mB�B�B + LG (5.1)

where eB > 0 and LG is a gauge-fixing term. We will work in the class of covariantgauges, so that

LG = − 1

2aB(∂µ Aµ

B)2

To begin, we consider the bare, regularized (finite) Green’s functions. It isimportant that there exist regularization procedures which preserve the gaugesymmetry of the theory. The ’t Hooft–Veltman dimensional regularization definedfor perturbative expansion (we can think, for instance, about perturbative expansionfor the bare Green’s function) belongs to this category. Thus, the results of

177

178 5 Quantum electrodynamics

our formal manipulations should always be understood as valid order-by-order inperturbation theory.

From the lagrangian (5.1) we can derive the Ward–Takahashi identities for thebare, regularized Green’s functions of our theory. Using these identities one canthen prove, allowing all parameters and fields to become renormalized, that thetheory is renormalizable without destroying its gauge invariance. Leaving asidethese formal considerations (consult, for example, Bogoliubov & Shirkov (1959),Slavnov & Faddeev (1980), Collins (1984)) we simply rewrite the lagrangian (5.1)in terms of the renormalized quantities assuming the most general form consistentwith gauge invariance after renormalization and we derive Ward identities for therenormalized Green’s functions (actually, using these Ward identities one can alsoprove a posteriori that the renormalized Green’s functions are finite).

The gauge-invariant structure of the QED lagrangian is not destroyed when eachgauge-invariant piece of it is multiplied by a constant. In addition the parametere can be arbitrarily changed. Thus the most general gauge-invariant form for thelagrangian (5.1) written in terms of the renormalized quantities reads†

L = − 14 Z3 Fµν Fµν+Z2�[i/∂−qe(Z1/Z2)A/ ]�−Z0m��−(1/2a)(∂µ Aµ)2 (5.2)

(as we shall see shortly, there is no need for a gauge-fixing counterterm). Therelations between bare and renormalized quantities are as follows

ABµ = Z1/2

3 Aµ, �B = Z1/22 �, eB = (Z1/Z2 Z1/2

3 )e

mB = (Z0/Z2)m = (m − δm)/Z2, aB = Z3a

Apart from the gauge-fixing term, lagrangian (5.2) is invariant under the gaugetransformations (1.91) and (1.94)

� ′(x) = exp[−iq�(x)]�(x)

A′µ(x) = Aµ(x)+ 1

e(Z2/Z1)∂µ�(x)

(5.3)

If a free scalar ghost field is included the full lagrangian is invariant under BRSsymmetry transformation (see Problem 3.3) on renormalized fields; the ghost field,being free, does not undergo renormalization. The renormalization constants Zi

must be fixed by imposing certain renormalization conditions. Our discussion inChapter 4 of the renormalization programme applies to the present case. In additionit is clear that Z1/Z2 must be finite as all other quantities in (5.3) are finite (makethis argument precise by considering BRS invariance and taking into account thatthe ghost field is not renormalized; compare with QCD, Section 8.1). ThereforeZ1 = Z2 up to finite terms. This result will be derived formally using the Ward–Takahashi identities. Actually, in most popular renormalization schemes Z1 = Z2

including finite terms.

† Strictly speaking one should include the scalar ghost field and refer to the BRS invariance (see Problem 3.3).

5.1 Ward–Takahashi identities 179

Mainly for historical reasons, the so-called on-shell renormalization scheme isthe one most commonly used in QED calculations. In this scheme we choose tointerpret the coupling constant e, the mass parameter m and the renormalized fieldswhich appear in the lagrangian (5.2) as the electric charge measured, for instance,in the Thomson scattering (see (12.86)), the physical mass of the fermion andthe ‘physical’ fields in the sense of Chapter 4, respectively. The renormalizationconditions are then imposed so as to ensure this interpretation of the renormalizedquantities in any order of perturbation theory. Generally speaking, this can beachieved by demanding that in the on-shell limit, when all the four-momentaapproach their on-shell values, in any order of perturbation expansion, the fermionand the photon propagator as well as the fermion–fermion–photon vertex functionapproach their zeroth order form. However, an infinite number of other renormal-ization prescriptions is possible. The renormalized quantities lose their previous in-terpretation but the theory can be formulated in terms of them in a totally equivalentway. In addition, in each order of perturbation theory there exist definite relationsbetween sets of parameters corresponding to different renormalization schemes.

5.1 Ward–Takahashi identities

General derivation by the functional technique

We derive the Ward–Takahashi identities in QED using the functional technique.The generating functional for the full Green’s functions is the following

W [J, α, α] =∫

D(Aµ��) exp(iSeff) (5.4)

where

Seff =∫

d4x [L(Aµ,�)+ Jµ Aµ + α� + �α] (5.5)

and the lagrangian L(Aµ,�) is given by (5.2). Under the functional integral onecan perform an infinitesimal gauge transformation

� ′ = [1 − iq�]�

A′µ = Aµ + 1

e∂µ�

(5.6)

where e = eZ1/Z2. The integration measure is invariant under infinitesimal gaugetransformation and the only gauge-dependent terms in the integrand are sourceterms and the gauge-fixing term. On the other hand, a change of the integrationvariables in (5.4) cannot change the value of the integral. Therefore we concludethat

δW/δ�(y)|θ=0 = 0 (5.7)


where

δW [J, α, α,�] = i∫

d4x∫

D(Aµ��) exp(iSeff)

×[

1

eJµ(x)∂

µ�(x)− iq�(x)α(x)�(x)+ iq�(x)�(x)α(x)

− 1

e(1/a)∂(x)

µ Aµ(x)∂2�(x)

](5.8)

Integrating expression (5.8) by parts, neglecting the surface terms at infinity andremembering that functional integrals of derivatives of fields are derivatives of theGreen’s functions (see Section 2.3) we get

1

a∂2∂µ

1

i

δW

δ Jµ(y)+ ∂µ Jµ(y)W + qeα(y)

δW

δα(y)+ qe

δW

δα(y)α(y) = 0 (5.9)

(remember that (→δ/δα)W = −W (

←δ/δα) or in terms of the generating functional

Z [J, α, α] for the connected Green’s functions (W = exp(iZ))

1

a∂2∂µ

δZ

δ Jµ(y)+ ∂µ Jµ(y)+ iqeα(y)

δZ

δα(y)+ iqe

δZ

δα(y)α(y) = 0 (5.10)

Eq. (5.10) is the general form of the Ward–Takahashi identities in QED, in a classof covariant gauges, for the connected Green’s functions. It is also very usefulto have the analogous relation for the 1PI Green’s functions. The generatingfunctional for the latter has been defined in Section 2.6. In the present case itsarguments are the ‘classical’ boson field Aµ(x) and the ‘classical’ fermion fields�(x) and �(x) (it is convenient to use the same notation for the arguments of � asfor the fields in the lagrangian; see Section 2.6) so that

�[Aµ(x),�(x), �(x)]

= Z [J, αα] −∫

d4x [Jµ(x)Aµ(x)+ α(x)�(x)+ �(x)α(x)] (5.11)

where (left derivatives only)

δZ

δ Jµ= Aµ, � = δZ

δα, � = −δZ

δα

Jµ = − δ�

δAµ, α = δ�

δ�, α = δ�

δ�

(5.12)

In terms of the functional �[Aµ,�, �] the identity (5.10) can be rewritten asfollows:

1

a∂2(y)∂

(y)µ Aµ(y)− ∂(y)

µ

δ�

δAµ(y)+ iqe

δ�

δ�(y)�(y)+ iqe�(y)

δ�

δ�(y)= 0

(5.13)


In the following we shall illustrate the general relations (5.10) and (5.13) withseveral examples.

Examples

As the first example we consider the Ward identity for the longitudinal part of theconnected photon propagator. Differentiating relation (5.10) with respect to Jν(z)and putting J = α = α = 0 one gets

−1

a∂2(y)∂

µ

(y)

δ2 Z

δ Jµ(y)δ Jν(z)

∣∣∣∣J=0

= ∂ν(y)δ(y − z) (5.14)

Using (2.134) and Fourier transforming into momentum space we obtain thefollowing relation:

i(1/a)k2kµGµν(k) = kν (5.15)

Here Gµν(k) is the Fourier transform of the connected part of the photon propaga-tor Gµν(x1 − x2). The solution to (5.15) reads

Gµν(k) = −ia(kµkν/k4)+ GTµν(k) (5.16)

where the transverse projection

GTµν(k) = (gµν − kµkν/k2) f (k2) (5.17)

gives zero contribution to the relation (5.15). The Ward identity (5.15) constrainsthe full renormalized propagator to the form (5.16). We observe that the longitu-dinal (gauge-dependent) part is not altered by interactions (compare with (3.32)).This means, among other things, that the renormalization programme does notrequire any counterterms for the gauge-fixing term.

Our next example is the well-known relation between the vertex and the fermionpropagators. Differentiating the general identity (5.10) with respect to

δ

δα(x)

δ

δα(z)= − δ

δα(z)

δ

δα(z)

and setting J = α = α = 0, one gets

1

a∂2(y)∂

(y)µ

δ2

δα(x)δα(z)

δ

δ Jµ(Y )Z

= iqeδ2 Z

δα(z)δα(y)δ(y − x)+ iqe

δ2

δα(x)δα(y)δ(y − x) (5.18)

Since

〈0|T�(x)�(y)|0〉 = iδ2 Z

δα(x)δα(y)

∣∣∣∣J=α=α=0


Fig. 5.1.

and

〈0|T Aµ(y)�(x)�(z)|0〉 = δ3 Z

δ Jµ(y)δα(x)δα(z)

∣∣∣∣J=α=α=0

relation (5.18) can be rewritten as follows

−(1/a)∂2(y)∂

µ

(y)〈0|T Aµ(y)�(x)�(z)|0〉= qe〈0|T�(y)�(z)|0〉δ(y − z)− qe〈0|T�(x)�(y)|0〉δ(y − z) (5.19)

This is the Ward identity in configuration space. Using translational invariance andFourier transforming into momentum space

〈0|T Aµ(y)�(x)�(z)|0〉=〈0|T Aµ(y − z)�(x − z)�(0)|0〉

=∫

d4 p

(2π)4

d4q

(2π)4exp[−ip(x − z)]

× exp[−iq(y − z)]Vµ(p, q)

〈0|T�(y − z)�(0)|0〉 =∫

d4k

(2π)4exp[−ik(y − z)]iS(k)

〈0|T�(x − y)�(0)|0〉 =∫

d4k

(2π)4exp[−ik(x − y)]iS(k)

(5.20)

then inserting (5.20) into (5.19) and integrating over∫

dx dy dz exp(ip′x) ×exp(iq ′y) exp(ik ′z) to get rid of the δ-functions, we obtain the Ward identity inmomentum space

−(1/a)q2qµVµ(p, q) = qeS(p + q)− qeS(p) (5.21)

Graphically, relation (5.19) can be represented by Fig. 5.1 and relation (5.21)in momentum space by Fig. 5.2, where the cross denotes −(1/a)∂2∂µ or−(1/a)q2qµ, respectively. Notice that the form (5.21) in momentum space canbe immediately deduced from the graphical representation of relation (5.19) inconfiguration space.


Fig. 5.2.

The Ward identity (5.21) can also be written in terms of the 1PI Green’sfunctions. We take the derivative of (5.13) with respect to

δ

δ�(x)

δ

δ�(z)

set Aµ = � = � = 0 and use the definitions (2.135) generalized to includefermions to get the following:

∂µ

(x)�(3)µ (y, x, z)− iqe�(2)(x, y)δ(y − z)+ iqe�(2)(y, z)δ(y − x) = 0 (5.22)

By analogy with (5.21) and (5.19), in momentum space relation (5.22) reads

qµ�(3)(p, q) = −iqe�(2)(p + q)+ iqe�(2)(p) (5.23)

where the �(n)s are Fourier transforms of the �(n)s. We recall (2.152): �(2) =i[G(2)(p)]−1 = +S−1(k) where, for example, for the free propagator, S(k) =1/(/k − m).

As an interesting consequence of the Ward identities (5.21) or (5.23) we noticethat the gauge parameter e = eZ1/Z2 and consequently the ratio Z1/Z2 mustbe finite if the theory is renormalizable. Indeed all other quantities, includinge, in these equations are renormalized quantities. In perturbative calculations,using a regularization procedure which preserves gauge invariance, the Zi s arepower series in the coupling constant α = e2/4π : (Zi − 1) = ∑∞

n=1 f (n)i αn .

The coefficients f (n)i contain pieces which are divergent in the limit where

regularization is absent (for example, when ε → 0 in dimensional regularization)and possibly also finite pieces. Thus the finiteness of the ratio Z1/Z2 implies thatorder-by-order in perturbation theory Z1 = Z2 up to the finite terms. In practice,in all the commonly used renormalization schemes Z1 = Z2 including the finiteterms. This is convenient (but not necessary) because the parameter e is thenuniversal for all fermions in the theory. It is also worth remembering that only inthe on-shell renormalization scheme is e the electric charge, i.e. e2/4π = 1/137.This point will be discussed in more detail in Section 6.4. In addition, whenZ1 = Z2 the counterterms form a gauge-invariant set and the Ward identities for


the renormalized Green’s functions are identical to those for the bare regularizedGreen’s functions. The latter can, of course, be derived in a similar way to therenormalized Ward identities. The only difference is that one must start with thelagrangian expressed in terms of the bare quantities.

5.2 Lowest order QED radiative corrections by the dimensionalregularization technique

General introduction

We recall the QED lagrangian (q = −1, e > 0) in a covariant gauge

L = − 14 Fµν Fµν + �(i/∂ + eA/)� − m�� − (1/2a)(∂µ Aµ)2 − (Z3 − 1) 1

4 Fµν Fµν

+ (Z2 − 1)�i/∂� − (Z0 − 1)�m� + (Z1 − 1)e�A/� (5.24)

(there is no counterterm to the gauge-fixing term: see Section 5.1). The Feynmanrules are given in Appendix B.

Dimensional regularization in QED in the presence of γ -matrices requires anextension of previous rules. We notice that, for example, in the relation

γµγν + γνγµ = 2gµν1l

the only space-time-dependent object is the tensor gµν . In n dimensions we canassume

Tr[γµγν] = 12 Tr[γµγν + γνγµ] = gµν Tr 1l = 4gµν (5.25)

i.e. we still consider γµs as 4 × 4 matrices in the fermion and antifermion spinspace, and define

gµµ = gµνgµν = n

This convention is the one most frequently used in calculations but other conven-tions are also possible (Collins 1984). In particular, in supersymmetric theoriesa dimensional regularization method called dimensional reduction (Siegel 1979)is used which preserves supersymmetry. This is not the case for the standarddimensional regularization convention used here. Several other relations andintegrals useful in the subsequent calculations are collected in Appendix B. Weshall calculate now all the elements which are necessary to discuss the lowestorder radiative corrections to QED processes such as, for example, the electronscattering (or e+e− pair production) in a static Coulomb potential. An importantexercise is to calculate the Thomson limit of the Compton scattering (Problem 5.2)in the on-shell and minimal subtraction renormalization schemes.

5.2 Lowest order QED radiative corrections 185

Fig. 5.3.

Vacuum polarization

We calculate the amplitude corresponding to the diagrams of Fig. 5.3. This is a one-loop correction to the free photon propagator. From consideration of Section 5.1we know that only the transverse part of the photon propagator has higher ordercorrections. On general grounds one can write it as follows:

i�µν(q) = iPµν�T(q2)

= i

(gµν − qµqν

q2

)q2�(q2) (5.26)

In the one-loop approximation the transverse part of the photon propagator reads

GTµν(q) =

−iPµν

q2+ (−iPµρ)

q2i�ρσ (q)

(−iPσν)

q2= −iPµν

q2[1 +�(q2)]

Summing over all one-particle reducible diagrams one gets

GTµν(q) =

−iPµν

q2[1 −�(q2)]

The scalar function �(q2) is given by

gµν�µν(q) = (n − 1)q2�(q2) (5.27)

where from the Feynman rules we obtain

i�µν(q) = −(ie)2∫

dn p

(2π)n

i

p2 − m2

i

(p − q)2 − m2Tr[γµ(/p + m)

× γν(/p − /q + m)] + i(Z3 − 1)(1)(qµqν − q2gµν) (5.28)

Therefore, after taking the trace

Tr[γµγαγνγβ] = 4(gµαgνβ + gµβgνα − gµνgαβ)

and using the Feynman parametrization 1/ab = ∫ 10 dx (1/[xa + (1 − x)b]2) one

gets

(n − 1)q2�(q2) = ie2∫

dn p

(2π)n

∫ 1

0dx

4(2 − n)(p2 − p · q)+ 4nm2

{x(p2 − m2)+ (1 − x)[(p − q)2 − m2]}2+ counterterm (5.29)


Using the integrals (B.18)–(B.20) we get finally

�(q2) = −α

π(4π)ε/2�

(12ε) ∫ 1

0dx 2x(1 − x)

[m2 − x(1 − x)q2 − iε

µ2

]−ε/2

= −α

π

[1

ε

2

3+ 1

3(ln 4π − γ )−

∫ 1

0dx 2x(1 − x) ln

m2 − x(1 − x)q2 − iε

µ2

]+ O(ε)− (Z3 − 1)(1) (5.30)

(we have taken n = 4−ε, α = e2/4π , and α is dimensionless in 4−ε dimensions:α → αµε, where µ is an arbitrary mass parameter). The pole in ε reflects the UVdivergence in four dimensions. It is cancelled by adding the Z3 counterterm. In theminimal subtraction scheme

(Z3 − 1)(1) = −α

π

2

3

1

ε(5.31)

and the finite part of the vacuum polarization, given by (5.30), can easily beevaluated in the small q2 and large q2 limits and reads (up to constant terms)†

α

3πln

m2

µ2− α

π

q2

m2

1

15+ O

(q4

m4

)q2 → 0

α

3πln|q2|µ2

|q2| � m2

(5.32)

The renormalization scheme most frequently used in QED is the on-shellrenormalization. In this case one demands:

�ON(q2) = 0 for q2 = 0

Therefore, from (5.30)

(Z3 − 1)(1) = −α

π

[2

3

1

ε+ 1

3(ln 4π − γ )− 13 ln

m2

µ2

](5.33)

and

�ON(q2) = α

π2∫ 1

0dx x(1 − x) ln

m2 − x(1 − x)q2 − iε

m2(5.34)

† The so-called MS scheme in which the counterterms include the constant terms: 2/ε → η = 2/ε− γ + ln 4πis also often used. It is useful to notice that in the MS, �(q2 = 0) = 0 for µ2 = m2. Hence αON = α(m2)

(see also Chapters 6 and 12). In the large q2 limit, in the MS scheme the sum over all one-loop one particlereducible diagrams gives

GTµν(q) =

−iPµν

q2(

1 − α3π ln |q2|

µ2

)This is the so-called leading logarithmic approximation for the photon propagator.


Fig. 5.4.

Asymptotically one obtains

�ON(q2)−→q2→0

−α

π

q2

m2

1

15+ O

(q4

m4

)�ON(q2) −→

−q2/m2→∞α

π

[13 ln

|q2|m2

− 5

9+ O

(m2

q2

)] (5.35)

As a final comment we rewrite �ON(q2) in the form of a dispersion integral (thedispersion technique will be discussed in some detail later). Firstly, we changevariables in the integral (5.34) into y = 1− 2x . Taking into account the symmetryy →−y of the integrand we can write

�ON(q2) = α

π

1

2

∫ 1

0dy

[∂

∂y(y − 1

3 y3)

]ln

[1 − q2(1 − y2)

4m2 − iε

](5.36)

and integrate it by parts to get

�ON(q2) = −α

π

1

2

∫ 1

0dy 2y(y − 1

3 y3)q2

4m2 − q2(1 − y2)− iε(5.37)

Another change of variable t = 4m2/(1 − y2) gives the result

�ON(q2) = −α

πq2∫ ∞

4m2

dt

t

1

t − q2 − iε

1

3

(1 + 2m2

t

)(1 − 4m2

t

)1/2

(5.38)

which is a so-called once subtracted dispersion relation.

Electron self-energy correction

We consider the electron self-energy truncated diagram and the necessary coun-terterms which are shown in Fig. 5.4. The proper two-point function �(2)( /p) reads

�(2)( /p) = /p − m −�(/p,m) (5.39)

The propagator is the inverse of the proper two-point function so we get

G(2)( /p) = i

/p − m −�(5.40)


The general structure of the electron self-energy correction is obviously thefollowing

�(/p,m) = �V (p2) /p +�m(p2)I (5.41)

It is often convenient to write it as an expansion

�(/p,m) = �0( /p = mF,m)+ ( /p − mF)�1( /p = mF,m)+ ( /p − mF)2�2( /p,m)

(5.42)where m2

F is the zero of the inverse propagator m2F = {[m + �m(mF)]/[1 −

�V (mF)]}2 to the order considered. We recall that �1 (actually 12�1: see

Section 4.1) appears as a correction to an external line of the on-shell scatteringamplitude. It can be calculated as (p2 = /p2)

�1 = ∂�

∂ /p

∣∣∣∣�p=mF

= ∂�m

∂p2

∣∣∣∣p2=mF

2mF +�V (m2F)+ 2m2

F

∂�V (p2)

∂p2

∣∣∣∣p2=m2

F

(5.43)

In this section we shall calculate �(/p) in the Feynman gauge (unlike �µν(q), theelectron self-energy is gauge-dependent). We can then find the one-loop relationbetween the pole mass mF and the renormalized mass parameter m as well as thecorrection �1. The calculation proceeds as follows. The Feynman integral reads.

−i�(/p) = (ie)2∫

dnk

(2π)nγ µ −igµν

k2 + iε

i

/p + /k − m + iεγ ν

− im(Z0 − 1)(1) + i(Z2 − 1)(1) /p (5.44)

As before we use dimensional regularization: n = 4 − ε. The functions �m(p2)

and �V (p2) are given by

Tr�(/p) = 4�m(p2)

Tr[/p�(/p)] = 4p2�V (p2)

}(5.45)

Introducing

N = γ µ(/p + /k + m)γ νgµν

we find (after some algebra)

Tr N = 4nm

Tr[/pN ] = 4[p2(2 − n)+ (p · k)(2 − n)]

}(5.46)

Next, we use Feynman parametrization for the denominators

1

k2

1

(p + k)2m2=∫ 1

0dx

1

[k2 + xp · 2k + x(p2 − m2)]2(5.47)


and (B.18)–(B.20) to perform the integration over momentum k to get the followingresults:

�m = nmC∫ 1

0dx X − (Z0 − 1)(1)m

�V = (2 − n)C∫ 1

0dx (1 − x)X − (Z2 − 1)(1)

(5.48)

where

C ≡ α

4π

(−1

4π

)−ε/2

�(

12ε)

X =[

x(p2 − m2)− x2 p2

µ2

]−ε/2

Using standard tricks to single out the pole terms in ε we immediately find therenormalization constants in the minimal subtraction scheme

(Z0 − 1)(1) = 2α

π

1

ε

(Z0 − 1)(1) = − α

2π

1

ε

(5.49)

The final (finite) expressions for �m and �V can then be easily written down, forexample,

�m(p2) = −α

πm

[∫dx ln

xm2 − x(1 − x)p2

4πµ2+ γ + 1

2

](5.50)

and similarly for �V (p2). We leave it to the reader to check that in the MS schemeat the one-loop level

mF = m(µ)

(1 + α(µ)

π

)We observe that �m and �V are IR finite (when p2 → m2). However, theirderivatives contributing to �1 are not. Indeed, differentiating the regularizedexpression (5.48) with respect to p2 one obtains the following result

∂�m

∂p2= nmC

(− 12ε) ∫ 1

0dx

x(1 − x)

µ2X1+2/ε (5.51)

which is, of course, UV finite (Zi are constants) and in the limit p2 → m2 gives(we can take m2

F = m2 because we work in the first order in α)

∂�m

∂p2

∣∣∣∣p2=m2

= n1

m

α

4π

(1 − γ

ε

2

)(m2

4πµ2

)−ε/2�(−ε)�(2)

�(2 − ε)(5.52)


The integration over Feynman parameters has introduced a new (IR) singularitywhich has again been regularized by keeping n = 4 − ε, but this time with ε < 0(changing ε from a negative to a positive value is legitimate since we are alreadydealing with a UV finite quantity). In order to keep trace of the origin of thesingular terms here and in the following we shall often distinguish between εUV

(n = 4 − εUV, εUV > 0) and εIR (n = 4 + εIR, εIR > 0).The final result reads (εIR > 0)

∂�m

∂p2

∣∣∣∣p2=m2

= 1

m

α

4π

[4

(1

εIR+ 1

2γ − 1 + 12 ln

m2

4πµ2

)+ 1

](5.53)

and for ∂�V /∂p2

∂�V

∂p2

∣∣∣∣p2=m2

= − 1

2m2

α

π

(1

εIR− 1 + 1

2γ + 12 ln

m2

4πµ2

)(5.54)

In addition

�V (p2 = m2) = α

4π

(ln

m2

4πµ2+ γ − 2

)(5.55)

so for �1, we get

�1 = ∂�

∂ /p

∣∣∣∣�p=m

= α

π

(1

εIR+ 3

4 lnm2

4πµ2− 1 + 3

4γ

)(5.56)

Our final remark is that although �V (p2) does not have IR singularities (whenp2 → m2), it nevertheless has the so-called mass singularities (when p2 → m2 →0).

Electron self-energy: IR singularities regularized by photon mass

The IR singularities encountered above are due to the zero mass of the photon.Instead of using the dimensional method one can regularize the IR singularities bygiving the photon a small mass λ, where eventually λ → 0 (actually this is themost standard method).

For the electron self-energy correction we now have

−i�(/p) = −e2∫

dnk

(2π)nγn( /p + /k + m)γ µ 1

k2 − λ2

1

(p + k)2 − m2


Fig. 5.5.

Following the previous calculation, instead of (5.50) one gets:

�m(p2) = −mα

π

∫dx ln

x2 p2 − x(p2 − m2 + λ2)+ λ2 − iε

4πµ2+ const.

�V (p2) = α

2π

∫dx (1 − x) ln

x2 p2 − x(p2 − m2 + λ2)+ λ2 − iε

4πµ2+ const.

(5.57)

These integrations can be done by writing the logarithm as ln p2(x − x1)(x − x2),where x1 and x2 are the roots of the quadratic expressions. The derivatives of �m

and �V can now be calculated in an elementary way. We get in particular

∂�m

∂p2

∣∣∣∣p2=m2

= mα

π

1

2m2ln

m2

λ2+ const. (5.58)

∂�V

∂p2

∣∣∣∣p2=m2

= α

π

(− 1

4m2

)ln

m2

λ2+ const. (5.59)

and finally

�1 = α

π

1

2ln

m2

λ2+ α

π

1

4ln

m2

4πµ2+ const. (5.60)

Comparing with the dimensional regularization of IR singularities we observe thecorrespondence 2/εIR → ln(m2/λ2).

On-shell vertex correction

We consider Fig. 5.5

�µ(p, p′) = (ie)2∫

dnk

(2π)nγα

i

/p′ + /k − mγµ

i

/p + /k − mγβ(−igαβ)

1

k2(Z1 − 1)(1)γµ

(5.61)


We work in the Feynman gauge and assume that the external electron lines areon-shell p′ 2 = p2 = m2. Using the identity γ µ/aγµ = (2 − n)/a, we can write theintegral as

�µ(p, p′)− (Z1 − 1)(1)γµ = −ie2∫

dnk

(2π)n

N

[(p′ + k)2 − m2][(p + k)2 − m2]k2

(5.62)

where

N = γµ[4p · p′ + (n − 2)k2 + 4(p + p′) · k]

− 4(p + p′)µ/k + kµ[4m − 2(n − 2)/k] (5.63)

We now introduce Feynman parametrization for the propagators and then have toperform the following integrations

2∫ 1

0dx dy y

∫dnk

(2π)n

[1, kα, kαkβ]

{k2 + [p′xy + p(1 − x)y]2k}3 (5.64)

Only the integral with two powers of k in the numerator is UV divergent (but IRfinite). We regularize it by taking n = 4−εUV. The other integrals are at most onlyIR divergent, so we take n = 4+ εIR there. To simplify notation we write the finalresults in terms of the integrals

I1 =∫ 1

0

dx

−q2x(1 − x)+ m2I3 =

∫ 1

0

dx x

−q2x(1 − x)+ m2

I2 =∫ 1

0

dx

−q2x(1 − x)+ m2ln−q2x(1 − x)+ m2

4πµ2

(5.65)

where q = p − p′ (we assume q2 ≤ 0) and µ2 is an arbitrary mass scale. We seethat I3 = 1

2 I1. We get finally

�µ = �UVµ +�IR

µ

where (in n = 4 − εUV dimensions)

�UVµ = γµ

α

π

[1

2εUV− 1

4(2 + γ )− 1

4

∫dx ln

−q2x(1 − x)+ m2

4πµ2

]+ α

2π

m

2(p + p′)µ I1 + γµ(Z1 − 1)(1) (5.66)

and (in n = 4 + εIR dimensions)

�IRµ = γµ

α

4π(2q2 − 4m2)

[(1

εIR+ 1

2γ

)I1 + 1

2 I2

]+ α

2πγµ(p + p′)2 I1

− α

2π(p + p′)µm I1 (5.66a)


The above result is often rewritten in terms of the Dirac and Pauli formfactors

�µ(p, p′) = γµF1(q2)+ 1

2(i/m)σµνqν F2(q2), σµν = 1

2 i[γµ, γν] (5.67)

Using the so-called Gordon decomposition

u(p′)γµu(p)− 12(i/m)u(p′)σµνqνu(p) = 1

2(1/m)u(p′)(p + p′)µu(p) (5.68)

and the relation (p + p′)2 = 4m2 − q2, one gets the following:

F1(q2) = α

π

[1

2εUV− 1

4(2 + γ )− 1

4

∫dx ln

−q2x(1 − x)+ m2

4πµ2+ m3 I3

]

+ (Z1 − 1)(1) + α

π

{(12q2 − m2

)[( 1

εIR+ 1

2γ

)I1 + 1

2 I2

]

+ (m2 − 12q2)I1

}(5.69)

F2(q2)

α

2πm2 I1 (5.70)

We can now make several observations. In the minimal subtraction scheme

(Z1 − 1)(1) = − α

2π

1

εUV(5.71)

and, comparing with (5.49), we have Z1 = Z2 to this order. One can check that forthe on-shell renormalization given by the conditions

∂�

∂ /p

∣∣∣∣�p=m

= 0; F1(q2 = 0) = 1

the relation Z1 = Z2 also holds. Indeed from (5.56) and (5.69) it follows that

Z1 = Z2 = 1 − α

π

(1

2εUV+ 3

4γ + 1 − 34 ln

m2

4πµ2− 1

εIR

)(5.72)

in this case (the renormalization constants now become IR divergent). At this pointwe encourage the reader to calculate the Thomson cross-section in the on-shellrenormalization scheme (Problem 5.2).

In our calculation we have reproduced the famous Schwinger result (Schwinger1949) for the anomalous magnetic moment of the electron

F2(q2 = 0) = α/2π (5.73)

Notice that the radiative correction (5.73) for the anomalous magnetic moment isfinite before renormalization. It depends on the renormalization scheme throughthe value of α(µ2) but that dependence is of the order O(α2). Correction (5.73) is


finite because the operator �σµν�Fµν has dimension five and cannot be presentin the original renormalizable lagrangian. It is generated by radiative correctionsas a non-local term which becomes local in the low energy limit q2 = 0 (see alsoChapter 12).

As the next point we discuss the structure of the leading logarithmic terms inour result for the vertex corrections in the limit |−q2| � m2. In the minimalsubtraction scheme we get

− α

4πln

−q2

4πµ2

as a finite contribution to F1 from �UV and†

α

π

[1

εIRln

m2

−q2− 1

4ln2 m2

−q2+( 1

2γ −1)

lnm2

−q2+ 1

2 ln−q2

4πµ2ln

m2

−q2

](5.74)

from �IR. Here the first two terms are the dimensional regularization form ofthe famous double logarithms due to the simultaneous presence of IR and masssingularities.

The IR singularities can also be regularized by λ, the photon mass. Astraightforward calculation of the leading double logarithms gives a result whichcorresponds to the replacement 2/εIR → ln(m2/λ2). As a final remark we noticethat the double logarithm terms do not depend on the renormalization scheme (Zi ,in the on-shell renormalization, for instance, contains only single logarithms).

5.3 Massless QED

For some high energy reactions it is useful to have results in the m = 0approximation. We recalculate now in the Feynman gauge the mass and vertexcorrections in this approximation. Consider first the self-energy correction. Wehave

−i�(/p) = −e2∫

dnk

(2π)nN

1

k2

1

(p + k)2(5.75)

where

N = (2 − n)( /p + /k)

† It is easy to see that

I1 ≈−1

−q22 ln

m2

−q2

I2 ≈−1

−q22 ln

−q2

4πµ2ln

m2

−q2+ 1

−q2

(ln2 −q2

m2+ 1

3π3)

O

(1

q4

)when |q2| � m2. In particular I2 can be written in terms of the Spence functions. The logarithms can also be

obtained simply by writing∫ 1

0 = ∫ ξ0 + ∫ 1

ξ and approximating the integrands in both integrals.

5.3 Massless QED 195

(remember γ µ/aγµ = (2− n)/a). Using Feynman parametrization and the integrals(B.18)–(B.20) one gets (n = 4 − ε and α → αµε)

�(/p) = α

π

1

4/p(2 − n)

(1

4π

)−ε/2(−p2

µ2

)−ε/2

�(

12ε) ∫ 1

0x−ε/2(1 − x)1−ε/2 dx

− (Z2 − 1)(1) /p ≡ �V (p2) /p (5.76)

We can now proceed in either of two ways. One is to keep external lines off-shellp2 < 0 to regularize the IR singularity. Then, in the minimal subtraction scheme

�(/p) = /p

(α

π

1

4ln

−p2

4πµ2+ const.

)(5.77)

The other way is to take external lines on-shell p2 = 0 and to use dimensionalregularization of the IR singularity. Note that for massless QED the finite wave-function renormalization �1 is given solely by �V (p2 = 0) and not its derivative.We rewrite (5.76) in the following form

�(/p) = α

4π(2 − n) /p

∫ ∞

−p2

dq2

q2

(q2

4πµ2

)−ε/2

�(1 + 1

2ε)

×∫ 1

0dx x−ε/2(1 − x)1−ε/2 − (Z2 − 1)(1) /p (5.78)

For p2 = 0 we split the integral into two:∫ µ2

0 + ∫∞µ2 which are IR and UV

divergent, respectively, when ε → 0 (to regularize the first integral we work inn = 4 + εIR dimensions, so we take ε → −εIR). Finally, after cancelling the UVdivergence by Z2, we get† in the minimal subtraction scheme

�V (p2 = 0) = − α

4π

2

εIR(5.79)

where the pole in εIR reflects the IR singularity.It is also straightforward to repeat the calculation of the vertex correction in the

massless electron case. We have

�µ = −ie2∫

dnk

(2π)n

N

(p′ + k)2(p + k)2k2(5.80)

where

N = γµ[4p · p′ + (n − 2)k2 + 4(p + p′) · k] − (p + p′)µ4/k − 2(n − 2)kµ/k

† This result is immediate if we note that∫∞µ2 −(Z2 − 1) is already UV finite and we can write it in (4 + εIR)

dimensions by changing ε → −εIR. Then the∫∞µ2 part cancels the

∫ µ2

0 part and the final result is simply

−(Z2 − 1) written in terms of εIR.


Repeating the steps which led us to the result (5.66), we now get (n = 4 − εUV)

�UVµ = γµ

α

2π�(1 + 1

2εUV)(2 − εUV)

(1

εUV− 1

)( −q2

4πµ2

)−εUV/2

×∫

dx dy y1−εUV[x(1 − x)]−εUV/2 + γµ(Z1 − 1)(1)

= γµ

α

2π

( −q2

4πµ2

)−εUV/2 �(1 − 1

2εUV)

�(1 − εUV)

1

εUV+ γµ(Z1 − 1)(1) (5.81)

and† (n = 4 + εIR)

�IRµ = γµ

(− α

2π

)( −q2

4πµ2

)εIR/2

�(1 − 1

2εIR)

×∫

dy yεIR−1(1 − y)∫

dx [x(1 − x)]εIR/2−1

= γµ

(− α

2π

)( −q2

4πµ2

)εIR/2 �(1 + 1

2εIR)

�(2 + εIR)

(2

εIR

)[1 + π2

6

(12εIR

)2]

(5.82)

The double pole structure in �IRµ is due to the simultaneous occurrence of IR

(massless photons) and mass (massless electrons) singularities. Both have beenregularized by working in n = 4 + εIR dimensions. We note also that, once(Z1 − 1)(1) is explicitly determined, for example, in the minimal subtractionscheme, we can rewrite �UV

µ (which is, of course, finite) in n = 4+εIR dimensionsby changing εUV →−εIR. Then the full vertex is written in (4 + εIR) dimensions.

5.4 Dispersion calculation of O(α) virtual corrections in massless QED, in(4 ∓ ε) dimensions

It is instructive to repeat the calculation of the electron self-energy and vertex cor-rections using dispersion relations. The dispersive technique has been extensivelyused in particle physics. We do not attempt its systematic discussion here, referringthe interested reader to other books (Bjorken & Drell 1965, Eden, Landshoff,Olive & Polkinghorne 1966). Our purpose is mainly a technical one: to show yetanother method of calculating Feynman diagrams which in some cases happens tobe instructive.

As we know, in 4 − ε dimensions Feynman amplitudes are UV convergent. Sowe can always write for them the dispersion relation without subtraction. However,the calculated amplitude, of course, has poles in ε and we have to perform next thestandard renormalization procedure, i.e. to add counterterms.

† �(1 − ε) ∼= �(1)− �′(1)ε + 12�

′′(1)ε2 = 1 + γ ε + 12 (γ

2 + 16π

2)ε2.

5.4 Dispersion calculation of O(α) virtual corrections 197

Fig. 5.6.

Self-energy calculation

The dispersion relation for the function �V (p2) defined by (5.41) reads

�V (q2 = 0) = 1

π

∫dp2

p2Im�V (p2) (5.83)

The imaginary part of a Feynman diagram amplitude is obtained by thereplacement

1

p2 − m2 + iε→−2π iδ(p2 − m2)�(p0) ≡ −2π iδ+(p2 − m2)

in the Feynman integral (Landau–Cutkosky rule). Therefore for the diagram shownin Fig. 5.6 we can write

Im�V (p2) = − 12 e2(2 − n)�(p2)

∫d�2 (1 − p · k/p2) (5.84)

where

d�2 = dnk

(2π)n2πδ+(k2)

dnl

(2π)n2πδ+(l2)(2π)nδ(n)(k + l − p)

= dnk

(2π)n2πδ+(k2)2πδ+((p − k)2) (5.85)

is the two-body phase space element in n dimensions. In the (k, l) centre of masssystem it can be written as follows in terms of momenta p = k+l and r = 1

2(k−l):

d�2 = 1

(2π)n−2dnr δ( 1

4 p2 + pr + r2)δ( 14 p2 − pr + r2)

= 1

(2π)n−2dnr 1

2δ(14 p2 + r2)δ(pr)

= 1

(2π)n−2dn−1r

1

2p0δ( 1

4 p2 − r2) (5.86)

We now introduce polar coordinates for the Euclidean vector r in n−1 dimensions

dn−1r = |r|n−2 d|r| d�n−1 = |r|n−2 d|r| dθ (sin θ)n−3 d�n−2 (5.87)


where 0 ≤ θ ≤ π and due to an azimuthal symmetry of the problem we integrateover all but one angle θ to get∫

dn−1r = (r2)(n−3)/2 dr2 (sin θ)n−3 dθπ(n−2)/2

�( 12(n − 2))

(5.88)

For the phase space element integrated over angles one then gets

d�2 = π1−ε/2

(2π)2−ε

1

�(1 − 12ε)

(p2

4

)−ε/2

(sin θ)n−3 dθ (5.89)

(we specify n = 4−ε and remember that r20 = 0 and p0 = (p2)

1/2 in cms). Finallyone can write

dθ (sin θ)n−3 = dz (1 − z2)−ε/2 = dy 2[4y(1 − y)]−ε/2

where z = cos θ and y = 12(1 + z) to get

d�2 = 1

8π

(p2

4π

)−ε/2 1

�(1 − 12ε)

dy [y(1 − y)]−ε/2 (5.90)

We are now ready to calculate Im�V (p2) and �V (q2 = 0). Since in cms p · k =12 p2 we obtain (as usual, in n = 4 − ε dimensions, we define dimensionless α byα → αµε)

�V (q2 = 0) = − α

4π

(2 − n)

�(1 − 12ε)

1

2

∫ ∞

0

dp2

p2

(p2

4πµ2

)−ε/2 ∫ 1

0dy [y(1 − y)]−ε/2

(5.91)which is equivalent to (5.78).

Vertex calculation

Calculation of the vertex correction proceeds similarly. We shall limit ourselvesto the UV finite part �IR

µ to understand better the origin of the IR and masssingularities. Writing

�IRµ = γµ2q2�(q2) (5.92)

and using the kinematical variables defined in Fig. 5.7 we have

Im�(q2) = −e2∫

d�2

[1 + 4k · (p − p′)

2q2

]1

k2(5.93)

5.4 Dispersion calculation of O(α) virtual corrections 199

Fig. 5.7.

(the term proportional to /k when integrated must give A /p + B /p′, so it does notcontribute to the on-shell limit), where

d�2 = dnk

(2π)n2πδ+((p + k)2)2πδ+((p′ + k)2)

= dnl

(2π)n2πδ+(l2)

dnl ′

(2π)n2πδ+(l ′ 2)(2π)nδ(n)(q + l + l ′) (5.94)

Therefore

Im�(q2) = −e2∫

d�2

(1

k2+ 2

q2

)(5.95)

The next step is to use the expression (5.90) for d�2. In the present case r =12(l − l ′) and in the (l, l ′) cms system (l+ l′ = p+ p′ = 0) r = l. Taking the z axisin the direction p we have

k2 = −2l · p = −2l0 p0 + 2l · p = −2( 12q0)

2 + 2( 12q0)z = − 1

2q2(1 − z)

where z = cos θ and finally (n = 4 + ε)

�(Q2) = α

4π

1

4πµ2

1

�(1 + 12ε)

∫ ∞

0

dq2

q2 − Q2

(q2

4πµ2

)ε/2−1

×∫ 1

0dy

(1

1 − y− 2

)[y(1 − y)]ε/2 (5.96)

The integral over dy gives [�(1+ 12ε)�( 1

2ε)/�(1+ 12ε)− �2(1+ 1

2ε)/�(2+ ε)]and it is divergent for ε → 0. The divergence comes from the region y ≈ 1 andtherefore z ≈ 1. In our specific reference frame z = 1 corresponds to l ‖ p, i.e. theexchanged photon has to have zero momentum. To integrate over q2 (for Q2 < 0)we change variables twice: q2 = −Q2x and u = 1/(1 + x). We then get the finalresult ∫ ∞

0

dq2

q2 − Q2(q2)ε/2−1 = (−Q2)ε/2−1�(1 − 1

2ε)�( 12ε) (5.97)


Fig. 5.8.

The pole in ε is due to the lower limit of integration over q2, i.e. due to a masslesselectron–positron intermediate state (mass singularity). Finally, using (5.92) weget (n = 4 + ε)

�IRµ = γµ

(− α

2π

)( −Q2

4πµ2

)ε/2 2

ε�(1 − 1

2ε)

[�(1 + 1

2ε)�( 12ε)

�(1 + ε)− �2(1 + 1

2ε)

�(2 + ε)

](5.98)

which can be seen to coincide with (5.82).

5.5 Coulomb scattering and the IR problem

Corrections of order α

In the Born approximation the scattering of an electron from a static Coulombpotential

A0(x) = Ze/4π |x|, Ai (x) = 0

is described by the diagram in Fig. 5.8. According to the general rules the crosssection reads

dσ0 = 1

2|p| |M(0)0 |22πδ(E ′ − E)

d3p′

(2π)32E ′ (5.99)

where

|M (0)0 |2 = 1

2

∑pol

|u(p′)γ 0u(p)|2 Z2e4

|q|4 (5.100)

and Ze2/|q|2 is the Fourier transform of the A0(x). We use the static approximationto the general formulae (see, for example, Halzen & Martin (1984)):

flux−1 = 1

2E p2EP |v − V| →1

2E p|v|1

2|p|

(2π)4δ(P + p − P ′ − p′)d3P′

2EP ′(2π)3

d3p′

2E p′(2π)3→ 2πδ(E ′ − E)

d3p′

2E p′(2π)3

5.5 Coulomb scattering and the IR problem 201

for P = P′ = 0 (remember, we use the relativistic normalization of 2E particles ina unit volume V = 1 – see Appendix A). As usual

12

∑pol

|u(p′)γ 0u(p)|2 = 12( /p

′ + m)αβ(γ0)βδ( /p + m)δγ (γ

0)γα = 12 Tr[· · ·]

(5.101)Performing the necessary calculations we find

dσ (0)0 = 1

4 Z2α2 1

|p|4 sin4( 12θ)/E2

[1 − β2 sin2( 12θ)] d� dE ′ δ(E − E ′) (5.102)

where β = |p|/E and p · p′ = 2|p|2 cos θ . We shall now study real and virtualcorrections of order α to the Born cross section.

Virtual corrections of order α are easily included using our results for theelectron self-energy, vacuum polarization and the vertex. The amplitude now reads

M (0)0 = u(p′)(1 + 1

2�1)[γ0 +�0(p, p′)](1 + 12�1)u(p)[1 +�(q2)]

≈ u(p′)γ0u(p)[1 +�1 + F1(q2)] (5.103)

In the final expression (5.103) we have neglected the terms O(α2) and thecorrections which are regular in the IR limit: �(q2) and the Pauli formfactorF2(q2). (In the amplitude for bremsstrahlung of a real photon we shall only beinterested in the limit of a soft photon.)

Including virtual corrections to order α we obtain therefore

dσ (1)0 = dσ (0)

0 [1 + 2�1 + 2F1(q2)] (5.104)

where using the results of Section 5.3

2�1 = 2α

π

(1

εIR+ 3

4ln

m2

4πµ2− 1 + 3

4γ

)(5.105)

(one-quarter (one-half) of the ln(m2/4πµ2) comes from the dimensional regular-ization of the UV (IR) divergence) and

2F1(q2) = −2

α

π

1

4ln

−q2

4πµ2− 2

α

π

[1

εIRln−q2

m2+ 1

4ln2 m2

−q2

+ ( 12γ − 1) ln

−q2

m2+ 1

2ln

−q2

4πµ2ln−q2

m2

](5.106)

The above expressions are valid for |q2| � m2 and in the minimal subtractionscheme. The first term in F1(q2) comes from the dimensional regularizationof the UV divergences. Together with the analogous term in �1 it gives(α/4π) ln(−q2/m2). The remaining µ2 dependence is due only to our methodof regularization of the IR singularity and should cancel out in every cross sectionfree of such singularities.


Fig. 5.9.

The origin of the IR singularity encountered in our cross section dσ (1)0 is in

the masslessness of the photon, or in other words in the long range nature ofthe electromagnetic interactions. In consequence the system has a high degree ofdegeneracy: a final state consisting of a single electron state is not distinguishableby any measurement from an e+n soft (k → 0) γ s state. Therefore the previouslydefined cross section dσ0 is not a physically meaningful (measurable) quantity.One feels that a summation over experimentally indistinguishable final states isnecessary. Following Bloch and Nordsieck and working to order α we introduce a‘measurable’ cross section σ as follows

σ =∫ E

0dE d

dE (σ(1)0 + σ

(1)1γ ) (5.107)

where E is the energy resolution of the detection equipment, E = E ′ − E is theenergy of the undetected real photon, E and E ′ are, as before, the energies of theinitial and final electrons in the laboratory system, and dσ1γ is the differential crosssection with a photon in the final state. We shall explicitly check that the crosssection σ is free of the IR singularity. Later we shall discuss this problem in moregeneral terms.

Working to order α we have to include dσ (1)0 given by (5.104) and dσ (1)

1γ whichis the cross section for a single photon emission given by the Feynman diagramsshown in Fig. 5.9. Using the standard rules we get the following:

dσ (1)1γ = |M (1)

1γ |21

|v|d3p′

(2π)3

d3k2k0(2π)3

2πδ(E − E ′ − k0) (5.108)

where in the soft photon approximation (momentum k neglected in the numeratorsof the Feynman expressions)

|M (1)1γ |2 = |M (0)

0 |2e2

[2p · p′

k · p k · p′− m2

(k · p)2− m2

(k · p′)2

](5.109)

In (5.109) the amplitude |M (0)0 |2 is defined by (5.100) and the sum over photon

polarizations has been taken. We observe that the corrections due to the soft photonemission factorize. The first term comes from the interference of the two diagrams


shown in Fig. 5.9, the next two are just the respective amplitudes squared. We nowproceed as follows∫ E

0dE

dσ (1)1γ

dE =∫ E

0dE

∫d3k

2k0(2π)32πδ(E − E ′ − k0)

E

|p||p′|2 d�

(2π)3|M (1)

1γ |2

∼=∫ E

0dE dσ (0)

0

dE

∫ |k|< E

0

d3k2|k|(2π)3

e2

×[

2p · p′

k · p k · p′− m2

(k · p)2− m2

(k · p′)2

](5.110)

where the last step is legitimate in the soft photon limit (we denote |k| = |k| = k0).At this point we get for the cross section, σ , the result

dσ

d�= dσ (0)

0

d�[1 + 2�1 + 2F1(q

2)+ I1 + I2 + I3] (5.111)

where, for example,

I1 = e2∫ |k|< E

0

d3k2|k|(2π)3

2p · p′

k · p k · p′(5.112)

and I2 and I3 are self-evident. Our last task is to calculate the corrections Ii . Theyare IR divergent and for consistency with the calculation of virtual corrections wemust regularize them by working in 4 + εIR dimensions. We replace

d3k2|k|(2π)3

→ dn−1k2|k|(2π)n−1

where |k| = (k21 + · · · + k2

n−1)1/2 and introduce spherical coordinates in n

dimensions (see (5.88)):∫dn−1k =

∫d|k| |k|n−2 sinn−3 θ1 sinn−4 θ2 . . . sin θn−3 dθ1 . . . dθn−2

= 2πn/2−1

�( 12 n − 1)

∫d|k| |k|n−2

∫ 1

−1dz (1 − z2)n/2−2 (5.113)

where z = cos θ1 and the integration over the remaining angles has been donein the last step (assuming the integrand depends only on θ1). Let us consider I2

first. Writing (k · p)2 = |k|2(E − |p|z)2 we can easily perform the necessaryintegrations. We see that the probability of a soft photon emission into an intervald|k| is proportional to d|k|/|k| which is easy to understand on dimensional grounds(soft emission does not depend on the electron momentum p). We see also that thedominant contribution to I2 comes from that region of phase space where k ‖ p.So, that part of the radiation tends to be collimated around the direction of motion


of the initial electron. The final result reads

I2 = −α

π

[1

εIR+ 1

2ln

( E)2

4πµ2− 1

2ln

4E2

m2+ const.

](5.114)

For I3 we get the same result with E changed into E ′. To calculate the term I1 it isconvenient to choose the reference frame in which p+ p′ = 0 (the Breit frame). Inthis frame

p · k = |k|(E − |p| cos θ)

p′ · k = |k|(E + |p| cos θ)

where the sign ˜ denotes quantities in the Breit frame and the condition |k| < Ein the laboratory system translates into |k| < E sin( 1

2θ), where θ is the scatteringangle in the laboratory (indeed pi · pf = p · p′ = 2E2 sin2( 1

2θ) in the laboratoryand pi · pf = 2E2 in the Breit frame; so E = E sin( 1

2θ)). In the Breit system theintegration over the angle can be performed with the help of Spence’s functionsand we get

I1∼= 2

α

π

[1

εIRln−q2

m2+ 1

2ln

( E)2

4πµ2ln−q2

m2+ 1

4ln2 m2

−q2+ 1

2γ ln−q2

m2+ f (θ)

](5.115)

for |q2| � m2. Adding all the contributions we obtain the final result

dσ

d�= dσ (0)

0

d�

{1 + α

π

[ln

−q2

( E)2+ ln

( E)2

−q2ln−q2

m2+ 3

2ln−q2

m2

+ f (θ)+ const.

]}(5.116)

(remember that −q2 ∼= 4E2). The cross section dσ/d� is indeed free ofthe IR singularity and depends on the experimental resolution E . The tran-sition probability decreases with decreasing E . Our calculation is valid for|(α/π) ln[−q2/( E)2]| � 1. Otherwise the correction becomes large and wemust include higher order contributions which, as we shall see later, exponentiate.The results presented here are for |q2| � m2. The limit |q2| � m2 can also beeasily obtained. It should be noted that although the final result (5.116) does not, ofcourse, depend on the method of regularization of the IR singularity, the splittinginto real and virtual contributions does. The coupling constant α is defined by theminimal subtraction renormalization scheme and α = αON[1 + O(αON)], whereαON = 1/137. So working to order α we can neglect the difference.

Finally we observe certain systematics in the cancellation of the IR singularitiesbetween virtual and real corrections. This can best be formulated if we represent


Fig. 5.10.

the cross section for real emissions by cut diagrams as shown in Fig. 5.10, wherethe factor 1

2 follows from the rule of Section 4.1 and the factor 2 takes account oftwo identical interference terms. The cancellation occurs between contributions tothe cross section represented by the diagrams in each column, separately.

IR problem to all orders in α

The previous results can be generalized to all orders in the electromagneticcoupling constant α (Yennie, Frautschi & Suura 1961). An important fact is thatin QED the IR divergent terms due to virtual corrections and to real soft emissionsexponentiate and cancel each other in the physically meaningful (observable) crosssection.

Let us consider a process in which there are some electrons and non-softphotons in the initial and final states and we represent their momenta by p and p′,respectively. Let the lowest order matrix element for our process be M (0)

0 (p, p′)and let M (n)

0 (p, p′) be the contribution corresponding to all diagrams in which thereare in addition n virtual photons. The complete matrix element is then

M0(p, p′) =∞∑

n=0

M (n)0 (p, p′) (5.117)


It has been shown (Yennie et al. 1961) that the M (n)s have the following structure

M (0)0 = m0

M (1)0 = m0αB + m1

M (2)0 = m0(αB)2/2 + m1αB + m2

...

M (n)0 =

n∑i=0

mn−i (αB)i/ i!

(5.118)

where m j is an IR finite function (independent of n) of order α j relative to M (0)0

and each factor αB contains the IR contribution from one virtual photon. Thus weget

M0 = exp(αB)

∞∑j=0

m j = exp(αB)M0 (5.119)

and the IR divergent exponent can be read off from our lowest order calculation,(5.104)–(5.106). An important intermediate step in proving (5.119) is to convinceoneself that IR divergences originate from only those virtual photons which haveboth ends terminating on external lines. Actually, this is to be expected onphysical grounds since long wave photons should not see charge distribution atshort distance.

For the Coulomb elastic scattering we get from (5.119) and (5.106), in the so-called double logarithmic approximation,

Mel = exp

(−α

πln−q2

m2ln−q2

λ2

)Mel (5.120)

where M is free of IR singularities. To write (5.120) we have changed thedimensional IR regularization into a cut-off by a small photon mass λ: 1/εIR →12 ln(m2/λ2) (see Section 5.2) in order to follow the traditional notation and in theexponent we have only retained terms quadratic in large logarithms, ln(−q2/m2)

and ln(−q2/λ2). The exponent in (5.120) is called the on-shell electron formfactorin the double logarithmic approximation. We observe that due to the IR divergencesencountered in order-by-order calculation the complete amplitude for purely elasticscattering is zero in the limit λ → 0. This is just a reflection of the fact that electronscattering is always accompanied by soft photon radiation and the cross section forelastic scattering is not a physical observable.

Let us now consider an inclusive cross section summed over any number ofundetected real photons in the final state, with total energy E such that E ≤ E ,where E is the energy resolution of the detectors. The differential cross section


for emission of m undetected real photons with total energy E reads

dσm

dE = exp(2αB)1

m!

∫ m∏i=1

d3ki

2ki0δ

(E −

∑i

ki0

)ρm(p, p′, k1 . . . km) (5.121)

where p and p′ represent, as before, momenta of all particles in the initial stateand of the detected particles in the final state. Virtual corrections to all orders inα are included in (5.121) and, as mentioned before, each σm is zero due to the IRdivergences. However, an inclusive cross section defined as

dσ

dE =∞∑

m=0

dσm

dE (5.122)

turns out to be free of IR singularities and thus it is a finite, physical quantity.Soft real photons must terminate exclusively on external lines in order to

contribute IR singularities. Writing (Yennie et al. 1961)

δ

(E −

∑i

ki0

)= 1

2π

∫ ∞

−∞dy exp

(iyE − iy

∑i

ki0

)(5.123)

using a generalization of (5.109)

ρm(p, p′, k1 . . . km) = σ0 p(k1) p(km) (5.124)

where p(k1) is given by the expression in the square brackets in (5.109) and, as inour order α calculation, neglecting for soft photon radiation the energy–momentumconservation constraints one gets the following result:

dσ

dE = exp[2α(B + B)]σ01

2π

∫ ∞

−∞dy exp[iyE + D(y)] (5.125)

where

2α B =∫ |k|<E d3ki

2ki0p(ki ) (5.126)

and

D =∫ |k|<E d3ki

2ki0p(ki )[exp(−iyki0)− 1] (5.127)

Thus all IR singularities due to real emissions are collected in the exponential factorexp(2α B), where B is given by our order α calculation, and they cancel with virtualIR singularities in B giving together (see (5.116)):

2α(B + B) = α

π

(ln−q2

E2+ ln

E2

−q2ln−q2

m2

)(5.128)


The cross section dσ/dE is finite, as expected. For a detector resolution E the integration over E up to E is necessary. One should also rememberthat the IR finite part σ0 contains, in general, terms like ln(q2/µ2), where q2

is some characteristic for the process four-momentum squared and µ2 is therenormalization point. The coupling constant α in (5.121) is α = α(µ2).

Problems

5.1 Derive the Ward identities (5.15) and (5.21) using the invariance of the QED Green’sfunctions under the BRS transformations (Problem 3.3). Hint: apply the BRStransformation to the Green’s functions

〈0|T Aµ(x)ω(y)|0〉 = 0

and

〈0|T�(x)�(y)ω(z)|0〉 = 0

where ω(x) is the free auxiliary scalar field.

5.2 Derive the Ward identity for the truncated connected Green’s function of the Comptonamplitude. For the forward Compton amplitude and in the limit of the photonmomentum k → 0 (the Thomson limit) express the amplitude in terms of the electronpropagator. Show that in the on-shell renormalization scheme the coupling constant ecan be defined by means of the Compton amplitude in the Thomson limit.

5.3 Study the singularity structure of the photon propagator in the one-loop approximationin the complex p2-plane and derive the Landau–Cutkosky rule. Hint: calculate thedifference P(p2+ i )− P(p2− i ), where P(p2) = p2�(p2) and �(p2) is definedby (5.26): choose the cms frame p = (p0, 0), study the pole structure of the integrandas a function of k0 and integrate over dk0 first.

5.4 Calculate the off-shell electron formfactor (Sudakov formfactor) in the double loga-rithmic approximation

�µ(p1, p2, q) = γµ exp

(− α

2πln|q2||p2

1|ln|q2||p2

2|

)|q2| � |p2

1|, |p22| � m2

5.5 Calculate the on-shell (see (5.120)) and the off-shell electron formfactors in the doublelogarithmic approximation using the light-like gauge n · A = 0 where n2 = 0.Compare with the calculation in the Feynman gauge.

6

Renormalization group

6.1 Renormalization group equation (RGE)

Derivation of the RGE

The spirit of the renormalization group approach lies in the observation discussedin Chapter 4 that in a specific theory the renormalized constants such as thecouplings or the masses are mathematical parameters which can be varied byarbitrarily changing the renormalization prescription. Once the infinities of thetheory have been subtracted out by a renormalization prescription R one is stillfree to perform further finite renormalizations resulting in each case in a differenteffective renormalization R′. Each renormalization prescription can be interpretedas a particular reordering of the perturbative expansion and expressing it in termsof the new renormalized constants. The latter are, of course, in each caserelated differently to physical constants (for example, the mass defined as thepole of the propagator is a physical constant) which are directly measurable andtherefore renormalization-invariant. A change in renormalization prescription iscompensated by simultaneous changes of the parameters of the theory so as toleave, by construction, all exact physical results renormalization-invariant. (Inpractice, there is a residual renormalization scheme dependence in each order ofperturbation theory.)

Most often, and also in this section, only subsets of arbitrary renormalizationprescription transformations which can be parametrized by a single mass scaleparameter µ are discussed. The parameter can be, for example, the value of thesubtraction point in the µ-subtraction schemes or the dimensionful parameter µ inthe minimal subtraction prescription. For a single-mass-scale-dependent subset ofrenormalization transformations we derive in the following a differential equationwhich controls the changes in the renormalized parameters induced by changingµ: the RGE. Actually, the set of renormalization prescription transformations {R}does not always have a group structure so the name is partly due to historicalreasons.

209

210 6 Renormalization group

Let us assume a particular renormalization scheme R and call �R the R-renormalized 1PI Green’s functions. Their relation to the bare Green’s functions�B will be

�R = Z(R)�B

where Z(R) denotes the appropriate product of renormalization constants definedwithin the R scheme. If we choose another renormalization scheme R′ then,

�R′ = Z(R′)�B

Obviously, there exists a relation between both renormalized Green’s functions,namely

�R′ = Z(R′, R)�R

where

Z(R′, R) = Z(R′)/Z(R)

Let us now consider the set of all possible Z(R′, R) for arbitrary R and R′. Amongthe elements of this set there is the composition law

Z(R′′, R) = Z(R′′, R′)Z(R′, R) (6.1)

and, in addition, to each element Z(R′, R) there is an inverse

Z−1(R′, R) = Z(R, R′)

and we can define the unit element as

Z(R, R) = 1

The composition law (6.1) is not always defined, however, for arbitrary pairs ofZs: the product

Z(Ri , R j )Z(Rk, Rl) (6.2)

is not, in general, an element of the set Z(R′, R), unless R j = Rk . A simpleexample of the above considerations is provided by the photon wave-functionrenormalization constant Z3 (de Rafael 1979).

Suppose we work in the µ-subtraction scheme and we want to relate tworenormalizations performed at subtraction points µ1 and µ2. The relevant quantitywill be Z(µ1, µ2) which in the one-electron loop approximation reads

Z(µ2, µ1) = 1 + α

2π

∫ 1

0dx 2x(1 − x) ln

µ22x(1 − x)+ m2

µ21x(1 − x)+ m2

(6.3)

This representation of Z obeys the composition law (6.1) but the group multiplica-tion law (6.2) is not obeyed, unless m = 0.

6.1 Renormalization group equation (RGE) 211

In this section we shall discuss the RGE in renormalization schemes such thatrenormalization constants Zi are mass-independent (mass-independent renormal-ization schemes). These include the minimal subtraction prescription and, ofcourse, in a trivial way any prescription for a massless theory.

The RGE in, for example, λ�4 theory can be derived as follows. We begin withthe relation between the renormalized and bare 1PI Green’s functions in the theoryregularized by the dimensional method, i.e. in 4 − ε dimensions

�(n)R (pi , λR,m R, µ, ε) = Zn/2

3 �(n)B (pi , λB,mB, ε) (6.4)

(for the definition of the renormalization constants Zi see Section 4.1). Thefinite limit (for ε → 0) of (6.4) exists since the theory is renormalizable. Thebare Green’s function �

(n)B (pi , λB,mB, ε) is in an obvious way independent of the

renormalization scale µ. The RGE will be derived by differentiating relation (6.4)with respect to the renormalization scale µ for fixed λB, dimensionful in 4 − ε

dimensions, mB and ε. In the mass-independent renormalization scheme we have

Zi = Zi (λR, ε)

where λR is defined as dimensionless, i.e.

λBµ−ε = Z−2

3 Z1λR

Similarly, we have

m2B = Z−1

3 Z0m2R

Differentiating (6.4) and using d�B/dµ = 0, one obtains

µ

(∂

∂µ+ dλR

dµ

∂

∂λR+ dm R

dµ

∂

∂m R

)�

(n)R (pi , λR,m R, µ, ε)

=(

12 nZn/2−1

3 µd

dµZ3

)�

(n)B (pi , λB,mB, ε)

=(

12 n

1

Z3µ

d

dµZ3

)�

(n)R (pi , λR,m R, µ, ε) (6.5)

We can define now the following functions:†

β(λR, ε) = µd

dµλR = λR

(µ

d

dµZ3

3 Z−11

)− ελR −→

ε→0β(λR) (6.6)

γ (λR, ε) = 12µ

d

dµln Z3 −→

ε→0γ (λR) (6.7)

γm(λR, ε) = µ

m R

dm R

dµ= 1

2µd

dµln(Z3 Z−1

0 )−→ε→0

γm(λR) (6.8)

† Similar considerations apply to additive renormalization.


The differential equations (6.6) and (6.8) simply express the µ-independence ofthe bare parameters λB and mB. Taking the limit ε → 0 in (6.5) (finite limits ofrelations (6.5)–(6.8) exist since the theory is renormalizable) we obtain the finalresult[µ

∂

∂µ+ β(λR)

∂

∂λR+ γm(λR)m R

∂

∂m R− nγ (λR)

]�

(n)R (pi , λR,m R, µ) = 0

(6.9)

It is important to remember that only in mass-independent renormalizationschemes or for m = 0 do the functions β, γm and γ depend on λR only. Ingeneral they also can depend on the ratio m R/µ. This happens, for instance, inthe µ-subtraction scheme.

Solving the RGE

Eq. (6.9) describes the change of renormalized parameters and multiplicativescaling factors for the renormalized Green’s functions which compensates a changeof the renormalization scale µ. To solve (6.9) suppose for a moment that γ (λR) ≡0. We then have a homogeneous equation µ(d/dµ)�

(n)R )(pi , λR,m R, µ) = 0 which

can be written in the form (subscript R in λR and m R will be omitted from now on):[− ∂

∂t+ β(λ)

∂

∂λ+ γm(λ)m

∂

∂m

]�

(n)R (pi , λ,m, µ0 exp(−t)) = 0 (6.10)

where µ = exp(−t)µ0, and µ0 is some fixed scale. We recall also that theparameters λ and m are renormalized at µ. As we shall see the meaning of thisequation is that the dependence of the function �

(n)R on t and hence on µ can be

expressed solely through the functions λ(t, λ), m(t, λ,m) defined by the equations∫ λ(t,λ)

λ

dx/β(x) = t

m(t, λ,m) = m exp

[∫ λ(t,λ)

λ

dx (γm(x)/β(x))

]= m exp

[∫ 1

0γm(λ(t

′)) dt ′]

(6.11)

Taking the partial derivative ∂/∂t one can rewrite (6.11) as differential equations

∂λ(t, λ)/∂t = β(λ)

∂m(t, λ,m)/∂t = γm(λ)m

}(6.12)

in accord with (6.6) and (6.8), with the boundary conditions

λ(0, λ) = λ, m(0, λ,m) = m


We also observe that (6.11) can be written in the form of the RGEs; for instance,differentiating λ(t, λ) given by the first equation of (6.11) with respect to λ onegets [

− ∂

∂t+ β(λ)

∂

∂λ

]λ(t, λ) = 0 (6.13)

Using (6.13) we can easily check that any function f (λ, t, µ0) = f (λ(λ, t), µ0) isµ-independent and satisfies the equation

µd

dµf =

(− ∂

∂t+ β(λ)

∂

∂λ

)f (λ(t, λ), µ0) = 0 (6.14)

Therefore the solution to (6.10) can be written as follows

�(n)R (pi , λ,m, µ0 exp(−t)︸︷︷︸

µ

) = �(n)R (pi , λ(t), m(t), µ0︸︷︷︸

µ exp(t)

) (6.15)

where λ(t) stands for λ(t, λ). The solution to the inhomogeneous (6.9) reads

�(n)R (pi , λ,m, µ) = �

(n)R (pi , λ(t), m(t), µ exp(t)) exp

[−n

∫ 1

0γ (λ(t ′)) dt ′

](6.16)

as can be checked by applying to it the differential operator of (6.9). Eq. (6.16)relates a Green’s function �

(n)R (pi , λ,m, µ), with the renormalized parameters

equal to λ and m at the renormalization scale µ, to the Green’s function�

(n)R (pi , λ(t), m(t), µ exp(t)), where λ(t) and m(t) can be understood as the

renormalized parameters corresponding to the scale µ exp(t).The renormalized parameters are renormalization-scale dependent. Any physi-

cal quantity P (renormalization-scheme-independent) can be calculated in terms ofthem: P = P(m, λ, µ). Being by definition renormalization-scheme-independentP must satisfy the homogeneous RGE:

µd

dµp =

[µ

∂

∂µ+ β(λ)

∂

∂λ+ γm(λ)m

∂

∂m

]P(λ,m, µ) ≡ RP(λ,m, µ) = 0

(6.17)

We shall now check, using (6.9), that this is indeed the case for the physical massdefined as the position of a pole in a propagator and for renormalized S-matrixelements. Let us consider the scalar propagator (p2) which satisfies the equation

[R+ 2γ (λ)] (p2, λ,m, µ) = 0 (6.18)

Assuming that has a pole at p2 = m2F, we can write a Laurent expansion for :

(p2, λ,m, µ) = R

p2 − m2F

+ (p2, λ,m, µ) (6.19)


The position of the pole, mF(λ,m, µ) and the residue of the pole R(λ,m, µ)

themselves satisfy RGEs which can be derived by using (6.19) in (6.18) andequating the residua of the poles at p2 = m2

F

Rm2F(m, λ, µ) = 0

[R+ 2γ (λ)]R(λ,m, µ) = 0

}(6.20)

We consider next an arbitrary n-scalar particle S-matrix element. This isobtained from the 1PI Green’s function �(n) by going to the physical poles p2

i = m2i

and multiplying by Rn/2 (see (4.27)). Since �(n) satisfies (6.9) and m2i and R satisfy

(6.20), we have

R limp2

i →m2i

Rn/2�(n) = limp2

i →m2i

RRn/2�(n) = limp2

i →m2i

[nγ (λ)− nγ (λ)]Rn/2�(n) = 0

(6.21)One must stress that it is an exact S-matrix element which is renormalization-scheme-independent. In perturbation theory truncated at order N there is always aresidual renormalization scheme dependence O(λN+1).

Green’s functions for rescaled momenta

The RGE is used most frequently in a somewhat different context, namely todiscuss the change in the Green’s functions when all momenta are rescaledpi → ρpi (but for fixed µ). Let us take a Green’s function of dimensionD : �(n)

R ∼ [M]D. On dimensional grounds it can be therefore written as follows

�(n)R (ρpi ,m, λ, µ) = µD f (ρ2 pi · p j/µ

2,m/µ, λ) (6.22)

We observe that �(n)R , is a homogeneous function of order D of variables m, ρ and

µ. Using the basic property of such functions we can write for it the followingequation: (

ρ∂

∂ρ+ m

∂

∂m+ µ

∂

∂µ− D

)�

(n)R (ρpi ,m, λ, µ) = 0 (6.23)

Combining (6.9) and (6.23) one gets (t = ln ρ)[− ∂

∂t+ β(λ)

∂

∂λ+ (γm − 1)m

∂

∂m− nγ (λ)+ D

]�

(n)R (ρpi ,m, λ, µ) = 0 (6.24)

which is the result we are looking for.We notice first that for m = 0 and for a non-interacting theory (β ≡ γ ≡ 0) the

solution to (6.24) reads

�(n)R (ρpi , λ, µ) = ρD�

(n)R (pi ,m, λ, µ) (6.25)


i.e. the Green’s functions exhibit canonical scaling (see Chapter 7). The generalsolution to (6.24) can be written using arguments similar to those used to solve(6.9) and it reads

�(n)R (exp(t)pi , λ,m, µ) = �

(n)R (pi , λ(t), m(t), µ) exp

[Dt − n

∫ t

0γ (λ(t ′)) dt ′

](6.26)

where

m(t) = m(t)/ρ

and λ(t) and m(t) satisfy (6.11). This solution can also be obtained directly by thefollowing steps

�(n)R (exp(t)pi , λ,m, µ) = �

(n)R (exp(t)pi , λ(t), m(t), exp(t)µ) exp

[−n∫γ (t ′) dt ′

]= �

(n)R (exp(t)pi , λ(t), exp(t) exp(−t)m(t), exp(t)µ)

× exp

(−n

∫γ dt ′

)= ρD�

(n)R (pi , λ(t), m(t), µ) exp

(−n

∫γ dt ′

)The first step corresponds to a change of the renormalization point according to(6.16) and the last step is a change of the mass scale by a factor ρ. We noticethat (6.26), unlike (6.16), relates Green’s functions which both have the samerenormalization scale µ, but different values of the renormalized couplings: λ,min the l.h.s. and λ, m(t) in the r.h.s. One can say, therefore, that they are Green’sfunctions of different theories.

Result (6.26) gives us Green’s functions for the rescaled momenta exp(t)pi (off-mass shell) in terms of the ‘effective’ parameters λ(t) and m(t) which are solutionsof (6.11). To use these results in physical applications one must know the functionsβ(λ), γ (λ) and γm(λ) which can only be calculated in perturbation theory.

Strictly speaking the RGE solution (6.26) applies in the deep Euclidean region,p2

i < 0, away from possible IR problems and new thresholds associated with time-like momenta.

RGE in QED

We choose to work in the set of mass-independent renormalization schemes andin a covariant gauge defined by the gauge-fixing term: −(1/a)(∂µ Aµ)2. The RGE


can then be derived following the steps outlined previously and we get[µ

∂

∂µ+ β(α, a)

∂

∂α+ γm(α, a)m

∂

∂m+ δ(α, a)

∂

∂a− nγ γγ (α, a)

− nFγF(α, a)

]�(nγ ,nF)(pi , α,m, µ, a) = 0 (6.27)

where in analogy to (6.6)–(6.8)

β(α, a) = limε→0

β(α, a, ε) = limε→0

µd

dµα

γm(α, a) = limε→0

µd

dµln Zm

γγ (α, a) = limε→0

12µ

d

dµln Z3

γF(α, a) = limε→0

12µ

d

dµln Z2

(6.28)

Derivatives are taken for fixed αB and ε, where α = e2/4π, αB = µε Z−13 α in

n = 4− ε dimensions, mB = Z−1m m, AµB = Z1/2

3 Aµ,�B = Z1/22 � (for definitions

of the renormalization constants, see Chapter 5). The only new element is theµ-dependence of the gauge-fixing parameter α = Z−1

3 aB which is responsible forthe term δ(α, a)∂/∂a with

δ(α, a) = limε→0

µda

dµ= lim

ε→0

(−aµ

d

dµln Z3

)(6.29)

The solution to (6.27) is obtained by the obvious modification of (6.16) with therunning parameters given by (6.11) and, in addition,

∂ a(t, α, a)∂t = δ(α, a), a(0, α) = a (6.30)

Using our previous experience we can also derive the RGE analogous to (6.26) forQED.

6.2 Calculation of the renormalization group functions β, γ, γm

We address ourselves to the problem of calculating functions, β(λ), γ (λ)

and γm(λ) in a mass-independent renormalization scheme. We recall thedefinitions (6.6)–(6.8), for example,

β(λ) = limε→0

β(λ, ε) = limε→0

µd

dµλ

∣∣∣∣fixed λB,ε

(6.31)

6.2 Calculation of renormalization group functions 217

In perturbation theory we can calculate the renormalization constants Zi whichrelate λ to λB

λB = µε Zλλ = µε

[1 +

∑ν

aν(λ)

εν

]λ (6.32)

where, for instance, Zλ = Z−23 Z1 for the λ�4 theory. The coefficients aν(λ) are

the series in λ

a1(λ) = a11λ+ a12λ2 + · · · + a1nλ

n

...

am(λ) = am1λ+ am2λ2 + · · · + amnλ

n

Using (6.6) we get

β(λ, ε) = −ελ− λZ−1λ µ

d

dµZλ (6.33)

Since

µd

dµZλ = µ

dλ

dµ

dZλ

dλ= β(λ, ε)

dZλ

dλ

we can rewrite (6.33) in the following form

β(λ, ε)Zλ + ελZλ + β(λ, ε)λ dZλ/dλ = 0 (6.34)

Substituting expansion (6.32) in (6.34) we look for the solution for β(λ, ε) in theform of a series β(λ, ε) = ∑

ν βνεν which is regular at ε = 0 (since our theory is

renormalizable a finite limit of β(λ, ε), for ε → 0, exists). Equating the residua ofthe poles in ε we obtain the following final result

β(λ, ε) = −ελ+ β(λ)

β(λ) = λ2 da1

dλ

λ2 daν+1

dλ= β(λ)

d

dλ(λaν)

(6.35)

We arrive, therefore, at two important conclusions

(i) β(λ) is totally determined by the residua of the simple poles in ε of Zλ.(ii) The residua of the higher order poles of Zλ are totally determined by a1(λ). This

reflects an important aspect of renormalization theory. In each successive order ofperturbation theory only one new divergence arises, i.e. the simple pole in ε. Therewill, of course, be multiple poles, up to 1/εN in the N th order but their residua aredetermined by lower order perturbation expansion. One can easily check in particularthat amn = 0 for m < n.


In a quite analogous way we also find the results for γ (λ) and γm(λ):

γ (λ) = − 12λ

d

dλZ (1)

3 (λ) (6.36)

where

Z3 = 1 +∑ν

Z (ν)

3

εν

and

γm(λ) = 12λ

dZ (1)m

dλ(6.37)

where

m2B = Zmm2, Zm = 1 +

∑ν

Z (ν)m

εν

(Zm = Z0 Z−13 in the λ�4 theory).

Using the results of Section 4.3 for the renormalization constants Zi in the λ�4

theory

Z1 = 1 + 3λ

16π2ε

Z0 = 1 + λ

16π2ε− λ2

2(16π2)ε+ 2λ2

(16π2)2ε2

Z3 = 1 − λ2

12(16π2)ε

(6.38)

and with help of (6.35), (6.36) and (6.37) one obtains the following renormalizationgroup functions in the λ�4 theory:

β(λ) = 3λ2

16π2+ O(λ3)

γm(λ) = 1

2

λ

16π2− 5

12

λ2

(16π2)2+ O(λ3)

γ (λ) = 1

12

(λ

16π2

)2

+ O(λ3)

(6.39)

For QED relation (6.32) is replaced by αB = µε Zαα with Zα = Z−13 . Using

(5.31) we get

Zα = 1 + α

π

2

3

1

ε+(α

π

)2

b2 + · · ·

6.3 Fixed points; effective coupling constant 219

and therefore, from (6.35)

β(α) = α

[α

π

2

3+(α

π

)2

b2 + · · ·]

(6.40)

Though not calculated explicitly, the second order contribution (α/π)2b2 has beensingled out for the sake of the discussion later. Note also that β(e) defined asβ(e) = µ(d/dµ)e is given by β(e) = 2πβ(α)/e. Therefore

β(e) = e3

16π2

4

3+ e5

(16π2)28b2 + · · · (6.41)

6.3 Fixed points; effective coupling constant

Fixed points

The main use of the renormalization group is in discussing the large, or small,momentum behaviour of a quantum field theory which is determined, as followsfrom (6.26) by the behaviour of the ‘effective’ coupling constant λ(t) as t →±∞.The latter is determined by ∫ λ(t,λ)

λ

dx/β(x) = t (6.11)

where β(λ) can be calculated perturbatively from (6.35). Eq. (6.11) provides abasis for a physically very important classification of different theories. We shalldiscuss here only theories with one coupling constant. First we assume (6.11) tobe valid in the whole range −∞ < t < ∞. Otherwise the renormalization scaleµ could not vary arbitrarily and the theory would need some natural cut-off �. Arenormalizable theory with perturbative expansion valid only in a certain energyrange cut-off by a scale parameter � (if, for example, new interactions switch on ata mass scale �) is not necessarily uninteresting physically and we shall come backto this point later.

The solution λ(t, λ) of (6.11) must for t →∞ and t → −∞ approach the zeroof β(x) nearest to λ or go to infinity if there is no zero to approach. A zero of β(x)is called a fixed point. Let λ be the fixed point nearest to λ. If

limt→∞ λ(t, λ) = λ ≡ λ+ (6.42)

it is called a UV stable fixed point; if

limt→−∞ λ(t, λ) = λ ≡ λ− (6.43)

it is called an IR stable fixed point.In both cases one says that λ is in the domain of attraction of the fixed point λ. A


Fig. 6.1.

domain is a region which lies between two zeros of β. Of course, for two differentvalues of the coupling λ lying in two different domains one, in general, has twodifferent theories (or two different phases of one theory). The nature of the fixedpoint is determined, if it is a simple zero, by the sign of the derivative β ′(λ) atλ = λ. From (6.11) and Fig. 6.1 it is clear that for

β ′(λ)|λ < 0, λ is a UV stable fixed point;

β ′(λ)|λ > 0, λ is an IR stable fixed point.

In other words, both for λ < λ and λ > λ, to satisfy (6.11) the effective couplingλ(t, λ) → λ for t → ∞ if β ′(λ)|λ < 0, and λ(t, λ) → λ for t → −∞ ifβ ′(λ)|λ > 0. (If λ is a multiple zero of β(λ), of odd order M , then the λ is a UV(IR) stable fixed point for β(λ)/(λ− λ)M < 0 (> 0).)

It is obvious but important to notice that for all single coupling theories the valueλ = 0 is a fixed point (β(0) = 0 in perturbation theory). A theory is said to beasymptotically free if λ = 0 is a UV stable fixed point and to be IR stable if it isan IR stable fixed point. All single coupling theories are either asymptotically freeor IR stable. In an asymptotically free theory the effective coupling vanishes forinfinite momenta. For theories with several couplings the origin may be UV stablefor some couplings and IR stable for others. The λ�4 theory (with λ ≥ 0) andQED are IR stable (β(λ) is positive for small λ) whereas QCD is asymptoticallyfree (β(λ) is negative for small λ). Of course, for a given theory, the only placethat we can compute β(λ) in perturbation theory is near λ = 0 and its behaviourfor increasing λ is essentially a speculation. Consequently, the behaviour of thecoupling constant λ(t, λ) for t →∞ in λ�4 and QED, and for t → −∞ in QCDalso remains, strictly speaking, a speculation. In Fig. 6.2 we show some examplesof what β(λ) might look like.


Fig. 6.2.

In Fig. 6.2(a) we have distinguished two domains of attraction. If λ is in region I,the theory is asymptotically free and for t → −∞ (low momentum) λ approachesthe fixed point λ−. On the other hand, if λ is in the region II then the theory isnot asymptotically free; instead λ(t, λ) → λ+ when t → ∞. Fig. 6.2(b) showswhat might happen in QED: IR stability and fixed point λ+ for t →∞. Fig. 6.2(c)shows another alternative for QED and Fig. 6.2(d) shows what we hope happens inQCD. In Fig. 6.2(d) there is no IR stable fixed point. To satisfy (6.11)∫ λ(−∞,λ)

λ

dx/β(x) = −∞

λ(t, λ) must diverge when t → −∞ and β(x) must be such that∫∞λ

dx/β(x) isdivergent. If the effective coupling becomes infinite in the IR region one oftenrefers to this situation as IR slavery. It is generally believed (but not proved) thatIR slavery in QCD is responsible for confinement.

Finally, suppose that the β-function is always positive (as in Fig. 6.2(c)) and thatit diverges at infinity in such a way that∫ ∞

λ

dx/β(x) = tc < ∞

It is likely that this situation occurs in the λ�4 theory (see (6.39)). For such a


theory the renormalized perturbative expansion cannot be consistent for all energies(unless λ(µ0) = 0 for any µ0). The theory must be modified by new interactionswhich switch on at some mass scale �c which provides a natural cut-off or mustundergo a phase transition at �c. For the theory to be valid up to �c the followingequation must be then satisfied∫ λ(�c,λ)

λ(µ0)

dx/β(x) = ln(�c, µ0) (6.44)

For a given �c the equation gives upper bounds for the low energy couplingconstants (assuming λ(�c, λ) = ∞) which, in turn, can be transformed into anupper bound for the number of elementary particles in the theory or their masses(see Problem 6.4).

Effective coupling constant

We will now study the explicit behaviour of the effective coupling constant inperturbation theory. With β(α), where α = λ for λ�4 and α = e2/4π for QEDand QCD, given by an expansion

β(α) = α[(α/π)b1 + (α/π)2b2 + (α/π)3b3 + · · ·] (6.45)

we can calculate α(t) integrating (6.12) order-by-order in α. We expand

α(t) =∑m=1

am(t)αm, a1 = 1 (6.46)

and

β(α(t)) =∑n=1

αn+1(t)bn/πn (6.47)

Therefore ∑m=2

dam(t)

dtαm =

∑n=1

bn

πn

[α +

∑m=2

am(t)αm

]n+1

(6.48)

Comparing the coefficients at the same powers of α and integrating over t one gets

a2(t) = b1t/π

a3(t) = (b1t/π)2 + b2t/π...

an(t) = (b1t/π)n−1 + O(tn−2) (6.49)

so that

α(t) = α

[1 +

∑n=1

(b1tα/π)n + O(tn−1αn)

](6.50)


For large t = 12 ln(µ2/µ2

0) or t = 12 ln(p2/p2

0), depending on whether we use(6.16) or (6.26), we can consider the so-called leading logarithm approximationin which we neglect terms O(tn−1αn) as compared to the terms O(tnαn). In thisapproximation the effective coupling constant reads

α(t) = α

1 − (α/π)b1t, α = α(t = 0) (6.51)

The same result can be obtained by expanding 1/β(x) in the integral (6.11) inpowers of x and directly integrating over x . In this way one gets the series

b1t =[π

α− π

α(t)

]+ B1 ln

α

α(t)+∑n=1

Cn

n

{[α(t)

π

]n

−(α

π

)n}(6.52)

where

Bn = bn+1/b1

and

Cn = (−1)n+1

∣∣∣∣∣∣∣∣∣∣∣

B1 B2 · · · Bn+1

1 B1 · · · Bn

0 1 B1 · · · Bn−1...

. . .. . .

...

0 0 0 1 B1

∣∣∣∣∣∣∣∣∣∣∣Truncating 1/β(x) at a given order in x gives us the corresponding approximatesolution for α(t). In the first approximation (b1 �= 0, bn = 0 for n > 1) we recoverthe leading-logarithm-result (6.51). In the second approximation (b1 �= 0, b2 �=0, bn = 0 for n > 2) we get the following

α(t) = α1(t)

[1 − α1(t)

π

b2

b1ln

(1 − α

πb1t

)](6.53)

where a1(t) is called the leading-logarithm-result (6.51).In QED b1 = + 2

3 (Section 5.2) and b2 = 12 (de Rafael & Rosner 1974). Taking

the limit t →−∞ in (6.53),

α(t) =t→−∞−

π

b1t+ πb2

b31t2

ln1

t+ O

(1

t2

)(6.54)

we see explicitly that the point α = 0 is an IR stable fixed point. In QCD b1 < 0and b2 < 0, therefore, α = 0 is a UV stable fixed point.

Another renormalization group parameter is the running mass m(t, µ, λ)

m(t,m, λ) = m exp

{∫ λ(t,λ)

λ

[γm(x)/β(x)] dx

}(6.55)


The behaviour of the Green’s functions for rescaled momenta exp(+t)pi iscontrolled by exp(−t)m(t). In particular, in a theory with the UV fixed point λ+one gets

exp(−t)m(t) = m exp{−t[1 − γm(λ+)]} exp

{−∫ λ(t)

λ

[dx/β(x)][γm(λ+)− γm(x)]

}(6.56)

Whether in the high energy limit masses can be neglected or not depends on γm(λ+)and on the integral in the exponent. For asymptotically free theories λ+ = 0 (andγm(0) = 0) so only the integral is responsible for the large t limit of m(t), and, forexample, for QCD, mass corrections are suppressed by exp(−t)t p as t →∞, withp > 0.

6.4 Renormalization scheme and gauge dependence of the RGE parameters

Renormalization scheme dependence

The last problem we would like to discuss in this chapter is the renormalization-scheme dependence within the class of mass-independent renormalization proce-dures and the gauge dependence of the perturbative results for the functions β, γm

and γ . The parameters of one renormalization scheme can be determined in termsof those in another scheme. If λ is the coupling in one scheme, another one willgive a coupling λ′

λ′ = F(λ) = λ+ O(λ2) for λ�4

= λ+ O(λ3) for trilinear coupling theories

}(6.57)

In the tree approximation the coupling is scheme-independent. The wave-functionrenormalization Z3 and mass renormalization Zm will be given in the new schemeby

Z ′m(λ

′) = Zm(λ)Fm(λ), Z ′3(λ

′) = Z3(λ)F3(λ) (6.58)

where Fi (λ) = 1 + O(λ2) (for the sake of definiteness we restrict our discussionto the case of the trilinear coupling theories (QED, QCD); modifications for λ�4

are trivial). For small enough λ we can assume that dF/dλ �= 0 and Fi �= 0. We

6.4 Renormalization scheme and gauge dependence 225

get therefore

β ′(λ′) = µd

dµ

∣∣∣∣λB,ε

λ′ = (dF/dλ)β(λ)

γ ′(λ′) = 12µ

d

dµ

∣∣∣∣λB,ε

ln Z ′3(λ

′) = γ (λ)+ 1

2

(d

dλln F3

)β(λ)

γ ′m(λ

′) = γm(λ)+ 1

2

(d

dλln Fm

)β(λ)

(6.59)

Using (6.59) the following results can easily be obtained (let λ and λ′ = F(λ) befixed points in both schemes)

(i)

dβ ′

dλ′

∣∣∣∣λ′= dβ

dλ

∣∣∣∣λ

(6.60)

(ii) If β(λ) = −2b0λ3 − 2b1λ

5 + · · · then

β ′(λ′) = −2b0λ′ 3 − 2b1λ

′ 5 + · · · (6.61)

i.e. the first two coefficients of the expansion for β are for mass-independentschemes, renormalization-scale-independent. Properties (i) and (ii) are importantto ensure that the nature of the fixed points (asymptotic freedom, IR stability) isscheme-independent.

(iii)

γ ′(λ′) = γ (λ), γ ′m(λ′) = γm(λ)

This again must be so since the values of γ and γm at a fixed point determine thescaling behaviour and masses, and have physical significance.

(iv) If γ (λ) = γ0λ2 + O(λ4) then γ ′(λ′) = γ0λ

′ 2 + O(λ′ 4) and similarly for γm(λ).

This is again important since γ0 and γ 0m determine the leading behaviour for t →

±∞ of the exponents in (6.26) and (6.11), respectively (scale factors for the fieldand the mass).

We stress that these properties hold only in the class of mass-independent renor-malization schemes. In particular within this class of renormalization schemesand in the leading logarithm approximation, α(q2) is scheme-independent. Inmass-dependent schemes the renormalization constants Zi and consequently thecoefficients bi , γi depend on the ratio m/µ and obviously cannot be equal to thosein the mass-independent schemes.


Effective α in QED

It is of some interest to find a link between the effective coupling constant in QEDand the ‘on-shell’ aON = 1/137 defined by (12.86). To this end we considerthe photon propagator renormalized in some mass-independent renormalizationscheme

αDµν(q) = −i

(gµν − qµqν

q2

)αd(q2/µ2, α)

q2− αai

qµqν

(q2)2(6.62)

where α = α(µ) and

d(q2/µ2, α) = 1

1 −�(q2/µ2, α)

The function � is the scalar factor of the vacuum polarization tensor �µν(q)defined in (5.26). An important observation is that in QED the quantity

αinv(q2) = αd(q2/µ2, α) = (α/Z3)Z3d(q2/µ2, α) = αBdB(q

2) (6.63)

is renormalization-scheme-independent because the bare quantities are. Inciden-tally, we see that in QED the product αBdB(q2) is finite. Relation (6.63) does nothold in QCD because then the Ward identities do not imply Z1 = Z2. Thus, in QCDthe relation between αd(q2, µ2, α) and αBdB(q2) involves uncancelled renormal-ization constants; αd(q2, µ2, α) is then renormalization-scheme-dependent andαBdB(q2) is not finite.

Turning again to QED we observe that

αON = limq2→0

αinv(q2) (6.64)

Indeed, in the on-shell renormalization scheme �ON(q2 = 0) = 0 and given thescheme independence of αinv defined by (6.63) we arrive at the result (6.64). Using(6.63) and (5.32) we also see that αON = α(m2), where α(m2) is defined in the MSscheme.

It remains to find the relation between the αinv(q2) and the running α(q2) definedas the solution to (6.11) We recall first that

α(q2) =q2→0

0

(it is an IR fixed point). Since the product α(µ)d(q2/µ2, α(µ)) is renormalization-scheme-independent we can change µ2 → q2 to get

αinv(q2) = α(q2)d(1, α(q2)) (6.65)

Using (5.32) for �(q2/µ2, α(µ2)) in the first order in α and with µ2 = q2 one gets

αinv(q2) =

q2→∞α(q2) (6.66)

6.4 Renormalization scheme and gauge dependence 227

This result actually holds to any order in α in the leading approximation because(see Problem 6.5)

�(q2/µ2, α(µ2)) = A1α(µ2) ln(|q2|/µ2)+

∑n=2

Anαn(µ2) lnn−1(|q2|/µ2)

+ non-leading terms

The value of αinv(q2) in the limit q2 = 0 can be obtained in terms of α(q2) fromknowledge of the first terms of the perturbation expansion again using (5.32) as thefinite limit of the expression

αinv(q2) =

q2→0

α(q2)

1 − [α(q2)/3π ] ln(|q2|/m2)= 0

0. (6.67)

The inverse relation to first order in αON = αinv(0) reads

α(q2) =q2→0

αON ln(|q2|/m2)�1

αON

(1 + αON

3πln|q2|m2

)(6.68)

Gauge dependence of the β-function

In QED the β(α)-function is gauge-independent. This follows immediately fromthe fact that the renormalization constant Z3 is gauge-independent. It renormalizesthe gauge-invariant operator Fµν Fµν or, in other words, the transverse part of thepropagator, which is gauge-invariant (see also Problem 6.6).

In non-abelian gauge theories the renormalization programme is more compli-cated (see Section 8.1) and β(α) is, in general, gauge-dependent. Nevertheless onecan prove that: (i) in any renormalization scheme the coefficient b1 in the expansion

β(α) = α

[b1

α

π+ b2

(α

π

)2

+ · · ·]

(6.69)

is gauge-independent, (ii) in the minimal subtraction scheme β(α) is gauge-independent.

To prove the first theorem we note that if we change the gauge we might changeα(µ) to α′(µ); both can be defined, for example, as the value of the three-pointfunction at some value µ2 of the kinematical invariants or according to the minimalsubtraction scheme. However, we can expand

α′(µ) = α(µ)+ O(α2) (6.70)

α′ = α in the zeroth order but the coefficients of the higher order terms mightdepend, in general, on the gauge parameter a. Now we consider the change of both


α′(µ) and α(µ) under a change of the renormalization point µ to calculate β ′(α′)and β(α)

β ′(α′) = µd

dµα′∣∣∣∣αB,aB,ε fixed

= µd

dµ[α(µ)+ O(α2)] = β(α)+ O(a3) (6.71)

because α(µ+ dµ) = α(µ)+ O(α2). This proves theorem (i).Theorem (ii) can be proved as follows: consider the bare coupling constant αB of

the theory. By its definition it is a fixed, gauge-independent parameter. Therefore

d

daαB

∣∣∣∣ε fixed

= daB

da

d

daBαB = 0 (6.72)

and using the relation αB = µε Zαα we have

0 = d

da(Zαα)

∣∣∣∣αB,ε fixed

= Zα

d

daα + α

d

daZα (6.73)

In the minimal subtraction scheme

Zα = 1 + Z (1)α (α, a)/ε + Z (2)

α (α, a)/ε2 + · · · (6.74)

and in particular in this scheme Zα does not contain any constant term C(α, a)which would destroy the following argument. Inserting the expansion (6.74) into(6.73) we get

dα

da+ 1

ε

(dα

daZ (1)

α + αdZ (1)

α

da

)+ O

(1

ε2

)= 0 (6.75)

The only way this equation can be satisfied is if dα/da = 0 and dZ (1)α /da = 0.

Thus in the minimal subtraction scheme the renormalized coupling constant α(µ)

is gauge-independent and consequently the same is true for the β(α) = µ(d/dµ)α.

Problems

6.1 Calculate the renormalization group functions β, γ, γm in QED in the momentum-subtraction renormalization scheme.

6.2 Consider a two-coupling theory: a non-abelian gauge theory with a scalar multiplettransforming according to the representation R of the gauge group. Write down thesystem of coupled RGEs for the gauge coupling g and for the meson self-couplingλ, to the lowest order in perturbation theory, and study the effect of the meson self-interaction on the asymptotic freedom of the theory. (See also Gross (1976).)

6.3 Consider the QED of nh ‘heavy’ fermions �h, each with non-vanishing mass M ,and nl ‘light’ fermions �l each with zero mass. Obtain the effective field theoryfor the light fields Aµ and �l, in the one-loop approximation by ‘integrating out’the heavy fermions from the Green’s functions with light field external legs: take the

Problems 229

photon propagator as an example and check the validity of the Appelquist–Carazzonedecoupling theorem in the on-shell renormalization scheme (Appelquist & Carazzone1975) and in the minimal subtraction scheme (Ovrut & Schnitzer 1981). In thelatter case understand the role of a finite renormalization of the coupling constantin absorbing the effect of the heavy mass. Study the RGE for the Green’s functions ofthe effective theory.

6.4 Consider the standard electroweak theory based on the gauge group SU (2) × U (1)with the gauge symmetry spontaneously broken (see Chapter 12) by one complexscalar doublet. Adopting the point of view expressed by (6.44) and using the lowestorder result for the β-function estimate the upper bound for the Higgs particle mass(Dasher & Neuberger 1983). Assume that for consistency the relation mH ≤ �c mustbe satisfied.

6.5 Using the RGE in a mass-independent renormalization scheme show that the vacuumpolarization in QED, (5.26), reads

�(q2/µ2, α(µ2)) = A1α(µ2) ln(|q2|/µ2)+

∑n=2

Anαn(µ2) lnn−1(|q2|/µ2)

+ non-leading terms

where the coefficients An are determined in terms of the b1 and b2 of the β-function.

6.6 Using the functional integral representation for the bare full two-photon Green’sfunction in QED, show by taking the derivative with respect to the gauge parameter athat the transverse part of the propagator is gauge-independent.

6.7 The model discussed in Problem 4.1 (taken now for m = M) has two couplingconstants. What symmetry does it possess when two couplings are equal? Discussthe fixed point structure for these couplings.

6.8 Consider λ�4 theory in the Euclidean space (Problem 4.3). Calculate the two-loopβ-function.

7

Scale invariance and operator product expansion

7.1 Scale invariance

Scale transformations

An important concept in quantum field theory is that of scale invariance. Amongother things, it is closely related to the famous Bjorken scaling in the deep inelasticlepton–hadron scattering. Scale transformation for the coordinates is defined asfollows

x → x ′ = exp(ε)x (7.1)

where ε is a real number. Obviously, all such transformations form an abeliangroup. Scale invariance is the invariance under the group of scale transformations.

If the configuration variables are scaled down as in (7.1) local fields �(x) (inthis section we denote by �(x) any scalar field ϕ(x) or spinor field �(x)) will be,in general, subject to a unitary transformation U (ε) (compare with Section 1.4)

U (ε)�(x)U−1(ε) = T−1(ε)�(eεx) (7.2)

where T (ε) is a finite-dimensional representation of the group (for classical fields�′(x ′) = T (ε)�(x)). We assume it to be fully reducible and therefore we canwrite

T (ε) = exp (−d�ε) (7.3)

where the constant d� is called the scale dimension of the field �(x).Let us consider first a free massless field theory. Its lagrangian does not contain

dimensionful constants and should be invariant under the transformation (7.2). Forfree fields one has the canonical equal-time commutation relations

[ϕ(x, t), ϕ(y, t)] = iδ(x − y) (7.4)

{�(x, t),�†(y, t)} = δ(x − y) (7.5)

and the scale dimension d� is then defined so that the commutation relations (7.4)

230

7.1 Scale invariance 231

and (7.5) remain invariant under the scale transformation. Multiplying rela-tions (7.4) and (7.5) by operators U and U−1 and using (7.2) one sees thatinvariance of (7.4) and (7.5) implies that these so-called canonical scale dimensionsare dϕ = 1 for scalar fields and d� = 3

2 for spinor fields. We see that the canonicalscale dimensions of fields coincide with their ordinary dimensions defined onpurely dimensional grounds (see Section 4.2).

In an infinitesimal form the scale transformation (7.2) is as follows:

U (ε)�(x)U−1(ε) = �(x)+ ε(d� + xµ(∂/∂xµ))�(x)+ O(ε2) (7.6)

For interacting field theories one still has the canonical commutation rela-tions (7.4) and (7.5) for bare fields. Defining scale dimensions for bare fields ascanonical ones, one can easily check that a lagrangian which does not containdimensionful constants is invariant under the scale transformation (7.2) (of course,strictly speaking, it is the action integral which is scale-invariant). Take, forinstance, the following lagrangian density

L = i�/∂� + 12(∂µϕ)(∂

µϕ)+ g�γ5�ϕ − (λ/4!)ϕ4 (7.7)

Using (7.6) and the values dϕ = 1, d� = 32 it is easy to calculate (see (1.29))

δL = ULU−1 − L

= L(exp(dφε)φ(x ′))− L(φ(x)) =∑

�=ϕ,�

∂L∂�

δ�+ ∂L∂(∂�/∂xµ)

δ

(∂�

∂xµ

)= ε(4 + xµ(∂/∂xµ))L(x) (7.8)

where we have used the relation

δ

(∂

∂xν

�(x)

)= ∂

∂xν

δφ(x) = ε

(d� + 1 + x · ∂

∂x

)(∂/∂xν)�(x)

This result is expected on general grounds since in a lagrangian which does notcontain dimensionful constants each term has a canonical dimension equal to itsordinary dimension which is four. Therefore the corresponding action integral isinvariant under scale transformations

δ I =∫

d4x δL(x) = 0 (7.9)

(one can also integrate explicitly using (7.8) and neglecting, as usual, the surfaceterms at infinity). Terms like m2�2(x), if added to (7.7), would, of course, breakthe scale invariance of the lagrangian as their scale dimension is two. Notice thedifference between scale dimension and ordinary dimension which is four for theterm m2�2; scale transformation is a transformation of dynamical variables (thefields) but not of the dimensionful parameters such as masses since we want to

232 7 Scale invariance and OPE

stay in the framework of the same physical theory. Terms like m2�2(x) break scaleinvariance explicitly. However, it is already worth recognizing at this point that inquantum field theory described by a scale-invariant lagrangian the scale invarianceis, in general, also broken. This is due to the necessity of renormalization whichintroduces a dimensionful parameter.

Turning back to an exactly scale-invariant world we can introduce the generatorD of the scale transformation

U (ε) = exp(iεD) (7.10)

and study its commutation relations with generators of the Poincare group (forconventions see Problem 7.1)

U (a) = exp(iaµPµ)

andU (ω) = exp(− 1

2 iωµν Mµν)

(7.11)

By comparing two sequences of dilatation and translation

�(x)→ε

exp(d�ε)�(exp(ε)x)→aµ

exp(d�ε)�(exp(ε)x + aµ)

and

�(x)→aµ

�(x + aµ)→ε

exp(d�ε)�(exp(ε)(x + aµ))

we conclude that

U (ε)U (a) = U (exp(ε)a)U (ε)

or

i[D, Pµ] = Pµ (7.12)

Similarly, one gets

U (ω)U (ε) = U (ε)U (ω)

and

[D, Mµν] = 0 (7.13)

From (7.6) we get the following differential representation of the generator D

i[D,�(x)] = (d� + xµ(∂/∂xµ))�(x) (7.14)


Dilatation current

The dilatation generator D can be written in terms of the Noether dilatationcurrent Dµ(x). The conserved dilatation current can be deduced by comparing aninfinitesimal change, obtained without the use of the Euler–Lagrange equations ofmotion, under the scale transformation of the lagrangian density of a scale-invarianttheory (7.8)

δL = ε

(4 + xµ

∂

∂xµ

)L = ε

∂

∂xµ

(xµL)

with an alternate expression for δL which does use Euler–Lagrange equations

δL = (∂/∂xµ)(�µδ�) � = (ϕ,�) (7.15)

where �µ is the conjugate momentum. The Noether dilatation current is thereforethe following

Dµ(x) = �µ(d� + xν(∂/∂xν))�(x)− xµL

or

Dµ(x) = �µd��(x)+ xν�canµν (7.16)

where

�canµν = �µ

∂

∂xν�(x)− gµνL

is the canonical energy–momentum tensor. The current is conserved in thepresence of the symmetry, but it may, of course, be defined even in the absenceof scale invariance.

The dilatation current can be written in a more compact way if we introduce amodified energy–momentum tensor �µν . Let us consider first a scalar field theoryand define �µν as follows:

�µν = �canµν − 1

6(∂µ∂ν − gµν�)ϕ2 (7.17)

The new tensor has the following properties

�µν = �νµ, ∂µ�µν = 0

and

Pµ =∫

d3x �can0µ =

∫d3x �0µ

Mµν =∫

d3x (xµ�0ν − xν�0µ)


i.e. �µν is a legitimate energy–momentum tensor of our theory. Using the relation�µ = ∂µϕ it is easy to check that in terms of �µν the Dµ of (7.16) (d� = 1) reads

Dµ = xν�µν − 16∂ν(xµ∂

ν − xν∂µ)ϕ2

The last term is the divergence of an antisymmetric tensor, so it does not contributeto ∂µDµ and to D. Thus we redefine

Dµ(x) = xν�µν(x) (7.18)

and

∂µDµ(x) = �µµ (7.19)

An important property of the new tensor is that its trace vanishes

�µµ = 0 (7.20)

for a scale-invariant scalar field theory (and

�µµ = m2ϕ2 (7.20a)

if a mass term is added) while the trace of �canµν contains singular derivative terms.

The new tensor was invented by Callan, Coleman & Jackiw (1970) when lookingfor an object with the ‘soft’ divergences (7.20a) in the context of searching foran energy–momentum tensor whose matrix elements between physical states arefinite. In a renormalizable field theory the finiteness of matrix elements of theT-ordered products of local currents is an additional requirement, which, in general,must be separately verified (see, for instance, Section 10.1). Actually, the proofof Callan, Coleman & Jackiw of the finiteness of matrix elements of the �µν isincomplete because it ignores the non-soft contribution (7.46) to the �µ

µ from theanomalous scale invariance breaking; see Adler, Collins & Duncan (1977) for thecomplete discussion.

Eq. (7.17) can be generalized to include fields with non-zero spin (Callan et al.1970):

�µν = Tµν − 16(∂µ∂ν − gµν�)ϕ2 (7.21)

where

Tµν = �µ∂ν�− gµνL+ 12∂

λ(�λ�µν�−�µ�λν�−�ν�λµ�) (7.22)


is the conventional symmetric energy–momentum tensor† and where all fields ofthe theory are assembled into a vector � whereas the scalar fields are denoted byϕ. The spin matrix �µν is defined by the transformation properties of the fieldsunder an infinitesimal Lorentz transformation

δ� = − 12(xµ∂ν�− xν∂µ�+�µν�)ωµν (7.23)

and reads, for example, �µν = 0 for spin 0 fields, �µν = − 12 iσµν = 1

4 [γµ, γν] forspin one-half fields. In terms of Tµν the generators of the Lorentz transformationare

Mµν =∫

d3x (xµT0ν − xνT0µ)

The tensor �µν defined by (7.21) has the same properties as the tensor (7.17)defined for scalar fields only. In particular its trace is a measure of scale invariancebreaking.

Conformal transformations

It turns out that many scale-invariant theories are invariant under a larger symmetrygroup of transformations, the so-called conformal group, which in addition totranslations, Lorentz transformations and scale transformation also contains specialconformal transformations. A conformal (i.e. angle-preserving) transformationmust leave invariant the ratio

dxα dyα|dx | |dy|

So it must have the property that

ds ′2 = gαβ dx ′α dx ′β = gαβ(∂x ′α/∂xγ )(∂x ′β/∂xδ) dxγ dxδ

= f (x) gαβ dxα dxβ ≡ f (x) ds2 (7.24)

where f (x) is a certain scalar function.For infinitesimal transformations

x ′µ = xµ + εµ(x)

† It can be constructed by noting that from conservation of the angular momentum current ∂λMλµν = 0, where

Mλµν = xµT λνcan − xνT λµ

can +�λ�µν�

and

Mµν =∫

d3x M0µν

we get

Tµνcan − T νµ

can + ∂λ(�λ�µν�) = 0

Eq. (7.22) follows if we require ∂µTµν = ∂νTµν = 0.


one therefore obtains the following condition

∂νεµ(x)+ ∂µεν(x) = h(x)gµν (7.25)

where h(x) = f (x)− 1. This equation gives

(n − 2)∂α∂βh(x) = 0

∂µ∂µεα(x) = 12(n − 2)∂αh(x)

where n is the dimension of the space-time. We see that for n > 2,† h(x) is at mostlinear in x and, consequently, εµ(x) is at most quadratic in x . The general solutionfor εµ(x) satisfying our constraints reads

εµ(x) = aµ + εxµ + ωµνxν + (−2b · xxµ + bµx2);ωµν = −ωνµ (7.26)

It includes (listed are the finite transformations)

translations x ′µ = xµ + aµ, h(x) = 0

scale transformation x ′µ = e+εxµ, h(x) = 2ε

Lorentz transformations x ′µ = �µν(ω)xν , h(x) = 0

special conformal transformations x ′µ = xµ + bµx2

1 − 2b · x + b2x2, h(x) = 4b · x

which in n dimensions form together the (n + 1 + n(n − 1)/2 + n)-parameterconformal group (see Problem 7.1). Conformal invariance implies scale invariancesince the commutator of an infinitesimal conformal transformation and an infinites-imal translation contains a scale transformation. However, the opposite is nottrue and examples of field theories which are scale-invariant but not conformallyinvariant can be constructed. Scale invariance implies conformal invariance if,and only if, there exists in the considered theory a symmetric traceless conservedenergy–momentum tensor so that ∂µDµ = �µ

µ = 0. It can be checked that sucha �µν exists in all renormalizable field theories of fields of spin ≤ 1. In this casethe currents

( j ε)µ(x) = �µν(x)εν(x) (7.27)

are conserved for any εν = εν(x, aα, ε, ωαβ, bα) satisfying (7.25) (check it) andindeed scale invariance goes together with the special conformal symmetry. From(7.26) and (7.27) we find explicitly the following four generators of the specialconformal transformations

K α =∫

d3x ( jα)0 =∫

d3x (x2�α0 − 2xαxν�ν0) (7.28)

For the transformation of fields under special conformal transformations seeProblem 7.1.† The case n = 2, interesting for statistical mechanics and string theories, requires a separate discussion; see

Problem 7.2.

7.2 Broken scale invariance 237

7.2 Broken scale invariance

General discussion

Scale invariance cannot be an exact symmetry of the real world. If it were, allparticles would have to be massless or their mass spectra continuous. Indeed, itfollows from commutation relation (7.12) that

[D, P2] = −2iP2

or, exponentiating,

exp(iεD)P2 exp(−iεD) = exp(2ε)P2 (7.29)

Acting on a single particle state |p〉 with four-momentum p(P2|p〉 = p2|p〉) onegets

P2 exp(−iεD)|p〉 = exp(2ε)p2 exp(−iεD)|p〉 (7.30)

i.e. the state exp(−iεD)|p〉 is an eigenstate of P2 with eigenvalue exp(2ε)p2. Ifwe assume in addition that the vacuum is unique under scale transformations (i.e.if scale invariance is not spontaneously broken)

exp(−iεD)|0〉 = |0〉then we conclude that

exp(−iεD)|p〉 = exp(−iεD)a†(p)|0〉 = exp(da)a†(eε p)|0〉 (7.31)

where a†(p) is the creation operator for the considered particle with momentump and da is its dimension. Result (7.31) means that the state exp(−iεD)|p〉 is aquantum of the same field as the state |p〉 but with a rescaled momentum (if thevacuum was not unique then the state exp(−iεD)|p〉 would belong to a differentHilbert space than the state |p〉 and our final conclusion would be avoided) andtherefore, by (7.30), all particles must be massless or the mass spectrum must becontinuous.

We will learn in the next chapters that there are several ways to account forthe symmetry breaking. The simplest idea for symmetry breaking is that of‘approximate’ symmetries. One assumes that there are terms in the lagrangian(e.g. mass terms in the case of scale invariance) which violate the symmetry,but that they are ‘small’. Another concept is that of spontaneous symmetrybreaking. Here the dynamical equations are completely symmetric, the Noethercurrents associated with the symmetry are conserved and the Ward identitiesare still valid, but the ground state is asymmetric. We believe this idea to berealized for chiral symmetry in strong interactions. Finally there is a third way:anomalous breaking of symmetries. Whenever a classical theory possesses asymmetry, but there is no way of quantizing the theory so as to preserve that


symmetry, we say that there are anomalies in the conservation equations forthe symmetry currents. More technically, quantum field theoretical calculationsmust be regularized to avoid infinities and anomalies occur when there does notexist a regularization procedure which respects the classical symmetry and thecorresponding Ward identity. Of course, a given symmetry may be broken byseveral of these mechanisms, simultaneously.

In quantum field theory, even with a scale-invariant lagrangian, the anomalousscale invariance breaking is quite expected. Scale invariance requires that therebe no dimensionful parameters whereas the regularization of the quantum theoryis effected by introducing a dimensionful cut-off or dimensionful coupling con-stants in the dimensional regularization procedure. Equivalently, an unavoidablerenormalization procedure necessarily introduces a scale at which the theory isrenormalized and this breaks scale invariance.

In the rest of this section we shall study the ‘anomalous’ response of thetheory to scale transformations by comparing the naive Ward identities with therenormalization group approach. For a discussion of the spontaneous breakingof scale invariance we refer the reader to the literature. In QCD the effects ofthe anomalous explicit breaking of scale symmetry are dominant, and there is noremaining consequence of the original classical scale symmetry.

Anomalous breaking of scale invariance

Ward identities for the dilatation current (trace identities) can be obtained from thegeneral formula (10.11)

∂

∂yµ〈0|T Dµ(y)�(x1) . . . �(xn)|0〉 = 〈0|T�µ

µ(y)�(x1) . . . �(xn)|0〉

− i∑

i

δ(xi − y)〈0|T�(x1) . . . δ�(xi ) . . . �(xn)|0〉 (7.32)

where δ�s are given by (7.14). By integrating with respect to variable y the l.h.s.vanishes: it gives surface terms which can be neglected if there are no zero massparticles coupled to �µ

µ; in reality the scale invariance may be spontaneouslybroken but simultaneously the corresponding Goldstone bosons (dilatons) getmasses due to anomalies; integrating with respect to y is equivalent to taking theFourier transform and letting k → 0. We then obtain∫

d4 y 〈0|T�µµ(y)�(x1) . . . �(xn)|0〉 = i

∑i

〈0|T�(x1) . . . δ�(xi ) . . . �(xn)|0〉

= i

(nd� + xµ

1

∂

∂xµ

1

+ · · · + xµn

∂

∂xµn

)〈0|T�(x1) . . . �(xn)|0〉 (7.33)


In the last step of (7.33) we have brought the derivatives outside the T -productwhich results in further equal-time commutator terms which, however, cancel inpairs. For example, for n = 2 we have

〈0|T�(x1)xµ

2

∂

∂xµ

2

�(x2)|0〉 + 〈0|T xµ

1

∂

∂xµ

1

�(x1)�(x2)|0〉

=(

xµ

1

∂

∂xµ

1

+ xµ

2

∂

∂xµ

2

)〈0|T�(x1)�(x2)|0〉 + x0

1δ(x01 − x0

2)〈0|[�(x1),�(x2)]|0〉

+ x02δ(x

01 − x0

2)〈0|[�(x2),�(x1)]|0〉 (7.34)

Defining the momentum-dependent Green’s functions

(2π)4δ

( n∑1

pi

)G(n)(p1, . . . , pn−1)

=∫

dx1 . . . dxn exp

(i∑

i

pi xi

)〈0|T�(x1) . . . �(xn)|0〉 (7.35)

and analogously for G(n)� (k, p1, . . . , pn−1) as the Fourier transform of the

〈0|T�µµ(y)�(x1) . . . �(xn)|0〉 we get from (7.33) the following Ward identity[

−n−1∑i=1

pi∂

∂pi+ n(d� − 4)+ 4

]G(n)(p1, . . . , pn−1) = −iG(n)

� (0, p1, . . . , pn−1)

(7.36)Summation over n−1 momenta and the additional term 4 in the square bracket aredue to the fact that, effectively, we have only n − 1 four-dimensional integrationsin (7.35). We observe that for d� = dcan the nd� − 4n + 4 is just the dimensionD of the Green’s function G(n)(p1, . . . , pn−1)(G(n) ∼ M D). In addition, if weparametrize all momenta by the scaling factor pi = ρ pi so that �i pi∂/∂pi =ρ∂/∂ρ we can rewrite (7.36) as follows (t = ln ρ)

[−∂/∂t + D]G(n)(exp(t)p1, . . . , exp(t)pn−1)

= −iG(n)� (0, exp(t)p1, . . . , exp(t)pn−1) (7.37)

From (7.37) one can derive analogous equations for the 1PI Green’s functions�(n)(p1, . . . , pn−1) and �

(n)� (0, p1, . . . , pn−1)

[−∂/∂t + D]�(n)(exp(t)pi ) = −i�(n)� (0, exp(t)pi ) (7.38)

where D = 4 − nd� is the dimension of the 1PI Green’s function �(n).Eq. (7.37) and (7.38) are the Ward identities following from the Noether relation

∂µDµ = �µµ and from the canonical field transformation rule (7.14). In particular,

for �µµ = 0, i.e. for a scale-invariant lagrangian, the solution to (7.38) reads

�(n)(ρpi ) = ρD�(n)(pi ) (7.39)


In this case the Green’s functions exhibit canonical scaling. However, we havealready mentioned that there is no reason to expect these identities to be valid inquantum field theory since the scale invariance is always broken by a dimensionfulcut-off. The response of a quantum field theory to the scale transformation is givenby the RGE (6.24).

Comparing (6.24) with the Ward identity (7.38) we see that the previouslymentioned anomalous breaking of canonical scale invariance occurs in quantumfield theory. Either the dilatation current has an anomalous divergence, or the fieldtransforms anomalously, or both. For easy reference we recall the solution to (6.24)

�(n)(exp(t)pi , λ,m, µ)

= exp(Dt)�(n)(pi , λ(t), m(t), µ) exp

[−n

∫ t

0γ (λ(t ′)) dt ′

](7.40)

where ∫ λ(t)

λ

dx/β(x) = t (7.41)

and

m(t) = (1/ρ)m exp

[∫ t

0γm(λ(t

′)) dt ′]

= (1/ρ)m exp

{∫ λ(t)

λ

dx [γm(x)/β(x)]

}(7.41a)

Consider first a massless m = 0 theory with a conserved Noether dilatationcurrent ∂µDµ = 0. As mentioned in Section 6.1 only for β(λ) = γ (λ) = 0, i.e.for a non-interacting theory, do we recover the canonical scaling, namely solution(7.39) to (7.38). Imagine now a theory with a UV fixed point λ+. The exponentialfactor in (7.40) can then be written as

exp

[−n

∫ t

0γ (λ(t ′)) dt ′

]= ρ−nγ (λ+)+ε(t)

where

ε(t) = −(1/t)∫ λ(t)

λ

dx n[γ (x)− γ (λ+)]/β(x) (7.42)

If the integral defining ε(t) is convergent, then ε(t) = O(1/t) and the theory isasymptotically, for t → ∞, scale-invariant. Apart from a trivial case when λ isexactly equal to λ+ and the RGE can then be solved directly and gives (7.39) withD → D − nγ (λ+), this happens when λ+ is a simple zero of β(x) : β(λ+) = 0,β ′(λ+) < 0 (we assume that γ (x) is differentiable). The asymptotic behaviour of


�(n)(exp(t)pi , λ, µ) is obtained by expanding λ(t) about λ+. The leading term is

�(n)(exp(t)pi , λ, µ) ∼ ρD−nγ (λ+)�(n)(pi , λ+, µ) (7.43)

The function γ (λ+) is called the anomalous dimension of the field. The lead-ing corrections to (7.43) arise from O(λ(t) − λ+) terms in the expansion of�(n)(pi , λ(t), µ) and of the exponential factor (7.42). More specifically one gets

�(n)(ρpi , λ, µ) = ρD−nγ (λ+) exp

{−∫ λ+

λ

dx [γ (x)− γ (λ+)]n/β(x)}

×�(n)(pi , λ+, µ)[1 + O(ρ−|β′(λ+)|)] (7.44)

Asymptotic scale invariance, in particular, is not a feature of asymptotically free(λ+ = 0) gauge theories. In this case β(x) = −bx3 + O(x5) and for most ofthe interesting operators γ (x) − γ (0) = 2cx2 + O(x4). So the integral (7.42) islogarithmically divergent and it causes logarithmic deviations from the asymptoticscale invariance (7.43) (remember that λ2(t) ∼ 1/2bt)

exp

(∫ λ(t)

λ

2c

b

dx

x

)∼ (2bλ2t)−c/b (7.45)

We also note an operator equation for the anomalous divergence of the dilatationcurrent in gauge theories reflecting the anomalous scale invariance breaking

�µµ = (β(λ)/2λ3)Ga

µνGµνa (7.46)

Rigorous proof of relation (7.46) is rather lengthy (Adler, Collins & Duncan 1977,Collins, Duncan & Joglekar 1977, Nielsen 1977). We see again that only for λ =λ+ �= 0 do we recover exact scale invariance, possibly with anomalous dimensions.

We now extend our discussion to theories with scale invariance explicitly brokenby mass terms. The identity (for bosons, to be specific)

m∂

∂m

i

p2 − m2= i

p2 − m2(−2im2)

i

p2 − m2(7.47)

implies that the operation m∂/∂m with external momenta and the coupling constantλ fixed (we assume that there is only one mass and one dimensionless couplingconstant) is equivalent to the insertion of a new vertex −im2�2 at zero-momentumtransfer. Thus we again see that for β = γm = γ = 0 (6.24) and (7.37) areidentical, with �µ

µ = m2�2.The mass effects in general are summarized in the solutions (7.40) and (7.41) to

the RGE. The ρ →∞ limit of (7.40) is controlled by the asymptotic behaviour of


both λ(t) and m(t) (we recall that t = ln ρ)

m(ρ) = mργm (λ+)−1 exp

{∫ λ

λ

dx [γm(x)− γm(λ+)]/β(x)}

−→ρ→∞ mργm (λ+)−1 × {logs of ρ} (7.48)

Now, for the mass effects to be asymptotically non-leading, it is essential that

m(ρ) −→ρ→∞ 0 or γm(λ+) < 1

Introducing the scale dimension d m of the mass operator m

d m ={

3 + γm(λ+) m = m��

2 + 2γm(λ+) m = �2m2

we get

d m < 4 (7.49)

as the condition for ‘softly’ broken scale invariance (Wilson 1969a). Otherwise themass insertion would effectively correspond to a hard operator and would influencethe asymptotic behaviour of the Green’s functions. Eq. (7.49) is automaticallysatisfied if the theory is asymptotically free because the relevant anomalousdimension γm vanishes.

7.3 Dimensional transmutation

The phenomenon of dimensional transmutation is closely related to scale invari-ance broken by the renormalization of a quantum field theory. To introduce thisconcept let us consider any dimensionless observable quantity A(Q) which candepend only on one dimensionful variable Q, for example, A(Q) = Q2 ·σtot. If weassume our theory to be described by a scale-invariant lagrangian (no dimensionfulparameters) then from dimensional analysis we must conclude that

A(Q) = const. (7.50)

Indeed, dimensional analysis tells us that, for example, a function f (x, y) whichdepends on two massive variables x, y and which is dimensionless and whosedefinition does not involve any massive constants must be a function of the ratiox/y only. So if it does not depend on y it must be a constant. As we know from ourexperience with quantum field theories the conclusion (7.50) is, in general, wrong.The reason why pure dimensional analysis breaks down in quantum field theory isthis: our quantity A actually also depends on the dimensionless free parameter(s)of the theory, for example, bare coupling constant(s): A = A(Q, gB). Moreover,the predictions of the theory cannot be expressed directly in terms of gB because

7.4 Operator product expansion (OPE) 243

the theory requires renormalization. The physics is independent of the details ofthe renormalization prescription but not of the necessity of renormalization whichproduces an effective scale. Dimensional transmutation is a process which exploitsthe above to introduce dimensionful parameters into the predictions of the theory.

An important illustration of these general considerations is provided by thebehaviour of the running coupling constant α(Q2) in QCD (we forget here abouta subtlety: α(Q2) is not really an observable; it depends on the definition of therenormalization scheme). One-parameter ambiguity for α(Q2) is reflected in thefact that the theory specifies the first derivative of α(Q2) rather than α(Q2) itself

dα/dt = β(α) (7.51)

where Q = exp(t)Q0, α = g2/4π and β(α) = (b1/π)a2 + O(α3). Integrating(7.51) we get

ln(Q/Q0) = F(α(Q))− F(α(Q0))

where

F(x) =∫

dx/β(x)

Thus, an effective scale Q0 appears as a constant of integration of the RGE (7.51).If we choose Q0 so that F(α(Q0)) = 0 and call it � then

α(Q) = F−1[ln(Q/�)] (7.52)

and in the leading logarithm approximation

α(Q) = −π/[b1 ln(Q/�)] (7.53)

�, defined by the condition F(α(�)) = 0, is the confinement scale: α(�) = ∞.Eq. (7.53) can be also rewritten in the following form

�/Q = exp{−π/[|b1|α(Q)]} (7.53a)

which tells us that given the strong coupling constant α(Q) at some Q the theorypredicts the confinement scale �. Dimensionless α(Q) and dimensionful � can betraded for each other.

7.4 Operator product expansion (OPE)

Short distance expansion

The OPE technique originates in studies of the relevance of scale invariance inquantum field theory (Wilson 1969a). In the previous section we have discussedhow mass terms and renormalizable interactions break this invariance which may,however, be re-established, at least up to logarithmic corrections, in the limit of


momenta going to infinity. In position space this corresponds to short distances,i.e. to vanishing separation for the operators in the matrix elements. OPE and scaleinvariance arguments allow us to determine the short distance singularity structureof operator products. The physical importance of studying operator products atshort distance will be discussed later. Here we shall introduce the technique of theOPE.

According to Wilson’s hypothesis a product of two local operators A(x) andB(y), when xµ is near to yµ, may be written as†

A(x)B(y) =∑

n

CnAB(x − y)On(

12(x + y)) (7.54)

where {On(x)} is a complete set of hermitean local normal-ordered operators and{Cn

AB(x)} is a set of c-number functions. The expansion is valid in the weak sense,i.e. when sandwiched between physical states. Relation (7.54) is true in free-fieldtheory (see Problem 7.4) and in any order of perturbation theory for interactingfields (Zimmermann 1970). Assuming scale invariance to be an approximatesymmetry at short distance one gets the leading behaviour of the Cs for smallvalues of their arguments, hopefully up to at most logarithmic corrections, interms of the scale dimensions of the operators. Indeed, commuting the dilatationgenerator D with (7.54) we get[

D, A(x)B(y)−∑

n

CnAB(x − y)On((x + y)/2)

]= 0 (7.55)

and in an exactly scale-invariant theory∑n

[dn − dA − dB − (x − y)

∂

∂(x − y)

]Cn

AB(x − y)On(12(x + y)) = 0 (7.56)

where dn , dA and dB are scale dimensions of the operators On , A and B,respectively. Since the operators On can be chosen to be independent we deducethat

(x − y)∂

∂(x − y)Cn

AB(x − y) = (dn − dA − dB)CnAB(x − y) (7.57)

and therefore CnAB is homogeneous of degree dn − dA − dB in its argument

CnAB(x) ∼ xdn−dA−dB (7.58)

Asymptotic scale invariance at short distance implies that relation (7.58) holdsat least in the limit x → 0. In QCD asymptotic scale invariance is broken bylogarithmic corrections (see Section 7.2) so relation (7.58) is modified but onlyby logarithmic terms. We see that the most singular contribution to A(x)B(y)

† The product also can be written in terms of, say, On(y) by expanding On(12 (x+ y)) in a Taylor series about y.


as x → y is given by the operator On having the lowest dimension. We havelearned in Section 7.2 that the scale dimensions dA, dB and dn are not, in general,equal to the canonical dimensions of the corresponding operators. Anomalousdimensions (see (7.43)) can be calculated in perturbation theory. However, forseveral interesting operators like conserved currents or currents of softly brokensymmetries, i.e. broken by operators with scale dimensions d < 4, the anomalousdimensions vanish (see Section 10.1).

The Lorentz transformation properties of the Wilson coefficients CnAB(x) require

them to be polynomials in the components of x times scalar functions of x2:Cn

AB = CnAB(x, x2). We have to determine prescriptions for handling light-cone

singularities of the coefficients CnAB for different types of products: time-ordered

products, simple products, commutators etc. To do this let us refer to free scalarfield theory. For instance, for the T -product of two currents J (x) = :�(x)�(x):in free scalar field theory we have (see Problem 7.4)

T J (x)J (0) = −2 2F(x,m2)I + 4i F(x,m2) :�(x)�(0): + :�2(x)�2(0):

=x → 0

−2 2F(x,m2)I + 4i F(x,m2)J (0)

+ 4i F(x,m2)xµ :�(0)(∂µ�)(0): + · · · (7.59)

where

F = −i〈0|T�(x)�(0)|0〉 = 1

(2π)4

∫d4k exp(−ikx)

1

k2 − m2 + iε

=x2→0

− i

4π2

1

x2 − iε(7.60)

is the free particle propagator in position space. Notice that :�(x)�(0): may beexpanded in a Taylor series since it has no singularity. Thus for T -products weconclude that scalar factors in Cs depend on x2 − iε. Since

T A(x)B(0) = A(x)B(0)�(x0)+ B(0)A(x)�(−x0)

and1

−x2 + iεx0�(x0)+ 1

−x2 − iεx0�(−x0) = − 1

x2 − iε(7.61)

for simple products we have the following prescription CnAB(x, x2) ≡ Cn

AB(x, x2−iεx0). For commutators [A(x), B(0)] we then get (1/(x±iε) = P(1/x)∓iπδ(x))†

1

x2 − iεx0− 1

x2 + iεx0= 2π iδ(x2)ε(x0) = ε(x0)2i Im

1

x2 − iε(7.62)

† Note in particular that:

= −i〈0|[�(x),�(0)]|0〉 = ε(x0)2 Im(i F) = −(1/2π)ε(x0)δ(x2)


and by differentiating both sides with respect to x2(1

−x2 + iεx0

)n

−(

1

−x2 − iεx0

)n

= − 2π i

(n − 1)!δ(n−1)(x2)ε(x0) (7.63)

where

ε(x) ={

1 for x > 0

−1 for x < 0

In a general case in which the light-cone singularity (7.63) is (x2)−p with somenon-integer power p, the difference

(−x2 + iεx0)−p − (−x2 − iεx0)

−p

appearing in a commutator [A(x), B(0)] vanishes for space-like separation x2 < 0:the prescription x2 → x2 − iεx0, by placing the cut along the positive real axis, isconsistent with causality.

Using the expansion of the form

[A(x), B(0)] =∑

n

CnAB On(0) (7.64)

where

CnAB = Cn

AB(x, x2 − iεx0)− CnAB(x, x2 + iεx0)

and letting x0 → 0 one can study the equal-time commutators. This is interestingbecause the appearance in the commutator of the so-called Schwinger terms(proportional to derivatives of the δ-functions) is linked to the singularity structureof operator products at short distances. Following Wilson (1969a) we can see thatthe coefficients Cn

AB are equivalent to a sum of δ-functions

CnAB = 0 Fn

AB(x0)δ(3)(x)+ 1Fn

AB(x0) · ∇∇∇δ(3)(x)+ · · · (7.65)

where

0 FnAB(x0) =

∫d3x Cn

AB(x0, x)

1FnAB(x0) = −

∫d3x Cn

AB(x0, x)x

etc. To get (7.65) one considers an integral∫d3x Cn

AB(x0, x)ρ(x) =∫

d3x CnAB(x0, x)(ρ(0)+ x · ∇ρ(0)+ · · ·) (7.66)

where ρ(x) is a differentiable trial function and the Taylor expansion is justifiedfor small x0 because Cn

AB vanishes for |x| ≥ |x0|. Eq. (7.65) follows immediatelyfrom (7.66) and from the definitions of the δ-function and its derivatives. The


dependence of the functions iFnAB(x0) on x0 is determined by dimensional analysis

to be

iFnAB(x0) ∼ x−2pn+3+i

0 (7.67)

when

CnAB ∼ x−2pn

The equal-time commutator is obtained in the limit x0 → 0. Thus for p > 1.5 thecoefficient of the δ-function is infinite and for p > 2 derivatives of the δ-functionsappear in the commutator. Some terms may vanish due to rotational invariance. Asan example let us take the product of two electromagnetic currents

Jµ(x) = :�(x)γµ�(x): (7.68)

In free fermion field theory one has (see Problem 7.4)

Jµ(x)Jv(0) ∼ x2gµν − 2xµxν

π4(x2 − iεx0)41l + O(x−3) (7.69)

The first term gives the familiar Schwinger term in the commutator [J0(x), Ji (0)](but not for two J0s):

[J0(x, 0), Ji (0)] ∼ Ci j∂ jδ(3)(x) (7.70)

where Ci j is quadratically divergent. We will see that the vacuum expectation valueof this term is related to the e+e− annihilation cross-section calculated under theassumption that the leading short distance singularities are those of the free-fieldtheory. Such terms are usually not revealed by direct calculation of the commutatorof two operators based on equal-time canonical commutation relations betweenfields and their conjugate momenta.

Light-cone expansion

The OPE can be extended to the light-cone region as well. It is obvious, however,that terms in (7.54) which are finite in the limit xµ → 0 may be singular in the limitx2 → 0 with xµ finite. Therefore we must expect an infinity of terms to contributeto the leading singularity on the light-cone, in contrast to the situation at x = 0where the leading behaviour is given by one term.

Let us look in more detail at the expansion of a product of two Lorentzscalar operators, for instance, two currents in free scalar field theory, (7.59). Asmentioned before, the Lorentz tensor operators (∂µ�)(0), (∂µ∂ν�)(0) etc. cannotbe neglected on the light-cone and the general structure of the expansion is as


follows:

A( 12 x)B(− 1

2 x) =∑

n

CnAB(x

2)xµ1 . . . xµ j Onµ1...µ j

(0) (7.71)

We have taken the bases Onµ1...µ j

to be symmetric traceless tensors with j Lorentzindices; they are irreducible tensors of spin j . We see that the behaviour of theexpansion for small x2 is determined by scalar functions of x2, the Cn

AB(x2) in the

neighbourhood of x = 0. (Note that we use the same notation for complete Wilsoncoefficients in (7.54) and for their scalar part in (7.71). This should not lead to anyconfusion.) As long as there are no extra divergences introduced by the summationover n, short distance behaviour and light-cone behaviour are connected

CnAB(x

2) = (x2)(dn− jn−dA−dB )/2 + higher orders in x2 (7.72)

so that the degree of singularity on the light-cone is determined by

τn = dn − jn (7.73)

The difference (dimension of an operator – its spin) is called twist. The leadingsingularity on the light-cone comes from operators with the lowest twist. Infree-field theory there exist infinite series of operators with fixed twist, sinceoperating on a field with a derivative raises the spin and the dimension by oneunit simultaneously. The scalar field � and the fermion field � as well as theirderivatives have twist one. Examples of twist two operators are

�∗ ↔∂µ1

↔∂µ2 . . .

↔∂µ j �

and

�γµ

↔∂µ1 . . .

↔∂µ j �

When dimensions are modified by the interactions it is not obvious that they arecorrelated with the spin in this way. Nevertheless, one usually takes the free-fieldtheory classification of operators as a guide in writing down the OPE.

As an exercise let us expand

�∗(− x

2

)�

(x

2

)= �∗(0)

[1 − ←

∂µ1

xµ1

2+ 1

2!←∂µ1

←∂µ2

xµ1

2

xµ2

2− · · ·

]×[

1 + xν1

2→∂ν1 +

1

2!

xν1

2

xν2

2→∂ν1

→∂ν2 + · · ·

]�(0)

=∑

n

1

n!

xµ1

2

xµ2

2· · · xµn

2�∗(0)

↔∂µ1

↔∂µ2 · · ·

↔∂µn�(0) (7.74)

Inserting (7.74) into (7.59) we get the leading terms in the light-cone expansionof the product of two scalar currents in free-field theory. In a similar way we canobtain the light-cone expansion for two electromagnetic currents.

7.5 The relevance of the light-cone 249

Fig. 7.1.

7.5 The relevance of the light-cone

Electron–positron annihilation

We briefly discuss now the relevance of the light-cone for high energy phenomena.Consider first the process of the electron-positron annihilation into hadrons. Itis a process widely investigated experimentally and interesting in many respectsas a laboratory for important discoveries in particle physics. We focus only onthe total cross section which is of the order of a few tens of nanobarns and iscomparable to the cross section for the e+e− → µ+µ−. In the one-photonexchange approximation (see Fig. 7.1) the amplitude is

Tn = e2v(p+)γµu(p−)(1/q2)〈n|Jµ(0)|0〉 (7.75)

where Jµ is the hadronic electromagnetic current. Thus the total cross section isdetermined by the tensor

Wµν(q) =∑

n

(2π)4δ(q − pn)〈0|Jµ(0)|n〉〈n|Jν(0)|0〉

=∫

d4x exp(iqx)〈0|Jµ(x)Jν(0)|0〉 (7.76)

Jµ is conserved and therefore one may write

Wµν(q) = ρ(q2)(q2gµν − qµqν) (7.77)

It can be checked that the total annihilation cross section for unpolarized initialleptons reads

σ(q2) = 8π2α2ρ(q2)/q2 (7.78)

To see what happens when q2 → ∞, we observe first that Wµν(q) can be writtenin terms of the current commutator

Wµν(q) =∫

d4x exp(iqx)〈0|[Jµ(x), Jν(0)]|0〉 (7.79)

Eq. (7.79) differs from (7.76) by the term

−(2π)4δ(q + pn)〈0|Jν(0)|n〉〈n|Jµ(0)|0〉


Fig. 7.2.

which is zero because q0 and p0n are positive. Since q2 > 0 we may choose a

frame in which q has only a time component and use the fact that the commutatorvanishes for x2 < 0. Then in this frame

Wµν(q) =∫ ∞

−∞dx0 exp[i(q2)1/2x0]

∫|x|<x0

d3x 〈0|[Jµ(x), Jν(0)]|0〉 (7.80)

Since the dominant contribution to the integral over x0 comes from regions withthe least rapid oscillations this Fourier transform is determined by the singularityof the commutator at x = 0. In a free-field theory the leading singularity of thecommutator originates from the term (7.69) which is responsible for the Schwingerterm (7.70). Using (7.63) and the relation∫

d4x exp(ikx)δ(n)(x2)ε(x0) = 22−2n

(n − 1)!π2i(k2)n−1�(k2)ε(k0) (7.81)

we easily find (for a single fermion with charge ±1)

Wµµ(q) =

q2→∞(1/2π)q2 (7.82)

and consequently

σ(q2) → 4πα2/3q2 (7.83)

This result is not surprising: a free-field theory is asymptotically scale-invariantand the cross section must behave like 1/q2 on dimensional grounds.

Deep inelastic hadron leptoproduction

Let us now consider another famous process: the production of hadrons inlepton–hadron collisions with very large momentum transfer. It is shown in Fig. 7.2which also contains some definitions of kinematical variables: k, k ′ are leptonfour-momenta; P is the four-momentum of the hadronic target (usually a proton);E, E ′ and θ are the lepton energies and lepton scattering angle in the laboratory


frame, respectively. As we shall see it is also very useful to define the so-calledBjorken variable

xBJ = −q2/2P · q (7.84)

For elastic scattering

s = m2 ⇒ 2P · q + q2 = 0 ⇒ xBJ = 1

In terms of the Bjorken variable, for fixed q2, the physical region for the process istherefore

0 < xBJ < 1 (7.85)

To be specific we consider the process of electroproduction. As in case ofthe e+e− annihilation into hadrons in the one-photon exchange approximation thecross section for this process is determined by the tensor

Wµν(P, q) =∑

n

(2π)4δ(q + P − pn)〈P|Jµ(0)|n〉〈n|Jν(0)|P〉

=∫

d4x exp(iqx)〈P|Jµ(x)Jν(0)|P〉

=∫

d4x exp(iqx)〈P|[Jµ(x), Jν(0)]|P〉 (7.86)

where Jµ is the hadron electromagnetic current. The passage from product tocommutator can be justified in a way similar to that for the annihilation reaction.We collect together some properties of the tensor Wµν(P, q), which are importantfor further discussion. If we introduce the forward Compton scattering amplitudefor photons with mass q2,

Tµν(q2, P · q) = i

∫d4x exp(iqx)〈P|T Jµ(x)Jν(0)|P〉 (7.87)

then it follows from the spectral representations (see also (7.62)) that

Wµν(q2, P · q) = 2 Disc Tµν(q

2, P · q) = 2 Im Tµν (7.88)

where Disc Tµν is the discontinuity of the Tµν along its branch cuts. The amplitudeTµν has normal threshold cuts in s = (P + q)2 and u = (P − q)2 for m2 < s < ∞and m2 < u < ∞. They correspond to a cut in xBJ for |xBJ| < 1. Note that onlyfor 0 < xBJ < 1 is the Wµν(q2, P · q) given by the deep inelastic cross section.

Now we are ready to discuss the relevance of the light-cone in our process. Letus choose a frame in which

P = (m, 0, 0, 0), q = (q0, 0, 0, q3)


and introduce the so-called light-cone variables

x± = x0 ± x3, x⊥ = (x1, x2)

q± = 12(q0 ± q3) = (ν/2m)[1 ± (1 − m2q2/ν2)1/2], ν = q0m

Then

Wµν(P, q) =∫

dx− exp(iq+x−)∫

dx+ exp(iq−x+)

×∫

x2⊥≤x+x−

d2x⊥ 〈P|[Jµ(x), Jν(0)]|P〉 (7.89)

where the limit on the x⊥-integration comes from the vanishing of the commutatorfor x2 = x+x− − x2

⊥ < 0. We see that in the limit

q+ → ∞q− fixed

}(7.90)

the behaviour of Wµν is determined by the behaviour of the integrand for x− →0, x+ finite, i.e. in the region with the least oscillations. Since the integrand isnon-vanishing only for x2

⊥ < x+x− (causality), x− → 0 for x+ finite correspondsto x2 → 0 (but not to xµ → 0) and Wµν is determined by the singularity of thecurrent commutator on the light-cone. In the limit (7.90) q2 → −∞, ν → ∞and xBJ is fixed (xBJ = −q2/2ν). The idea of studying the limit (7.90) is due toBjorken (1969) and it has turned out to be one of the most creative ideas in modernelementary particle physics.

For simplicity let us study the scalar current analogue of the electroproductiontensor

W (q2, P · q) =∫

d4x exp(iqx)〈P|[J (x), J (0)]|P〉 (7.91)

in the Bjorken limit. Spin complications are not essential for general orientation.Apart from the c-number contribution which is irrelevant for the considered matrixelement (first term in (7.59)) the leading singularities of the operator products onthe light-cone are given by the twist two operators

〈P|[J (x), J (0)]|P〉 = 1

π22π iδ(x2)ε(x0)

∑n

1

n!xµ1 . . . xµn

×〈P|�(0)∂µ1 . . . ∂µn�(0)|P〉 + less singular terms

(7.92)

where we have used (7.63) and (7.74). For the realistic case of particles with spin,twist two operators are again the leading ones. The general form of the matrix


element in (7.92) is

〈P|Oµ1...µn |P〉 = An Pµ1 . . . Pµn + δµ1µ2 Pµ3 . . . Pµn + · · · (7.93)

Terms having at least one Kronecker delta may be ignored because when contractedwith the xµs in (7.92) they give additional factors x2 and therefore are less singular.So we have

〈P|[J (x), J (0)]|P〉 = 2i

πδ(x2)ε(x0) f (x P)+ less singular terms (7.94)

where

f (x P) =∑

n

1

n!(x · P)n An

Defining the Fourier transform

f (x P) = (1/2π)

∫dξ exp(iξ x · P) f (ξ)

and using the identity†

i∫

d4x exp(ikx)δ(x2)ε(x0) = (2π)2ε(k0)δ(k2) (7.95)

we get

W (q2, P · q) = 4∫

dξ δ((ξ P + q)2)ε(q0 + ξ P0) f (ξ)

The roots of the argument of the δ-function are

ξ± = − ν

m2± ν

m2

(1 − q2m2

ν2

)1/2

−→BJ

−q2

2ν,− 2ν

m2(7.96)

where ‘BJ’ means Bjorken limit. As has already been explained, W (q2, P · q),being the imaginary part of the amplitude for forward off-mass-shell Comptonscattering, is non-vanishing for |−q2/2ν| < 1. Thus only the ξ+ is relevant and interms of the Bjorken variable our result reads

νW (q2, P · q) = 2 f (xBJ) (7.97)

In the free-field theory with twist two operators giving the leading singularity on

† Identity (7.95) follows from the spectral representation for the free-field commutator (Bjorken & Drell 1965)and the relation (7.62):

〈0|[�(x),�(0)]|0〉 = − 1

(2π)3

∫d4k exp(ikx)δ(k2 − m2)ε(k0)

=(

i

2π

)δ(x2)ε(x0)


the light-cone, we obtain the famous Bjorken scaling for the structure functionW (q2, P · q).

An important final observation is that only even spin operators are allowed in theexpansion (7.92). Indeed W is real (it is a cross section) so f (ξ) must be real andtherefore f (x P) must be even in x .

Wilson coefficients and moments of the structure function

Use of the OPE is by no means limited to the free-field theory case. Its validityfor renormalized operators has been proved in perturbation theory and therefore itcan be used to obtain further information about the scaling limit of the structurefunctions, beyond that contained in the hypothesis of canonical dimensions. Animportant step in this direction is to establish some relation between the Wilsoncoefficients and measurable quantities which is not limited to the canonical scalingcase. The standard procedure is as follows. Consider first the forward scatteringamplitude for scalar currents

T (q2, P · q) = i∫

d4x exp(iqx)〈P|T J ( 12 x)J (− 1

2 x)|P〉 (7.98)

The structure function (7.91) is the discontinuity of T , see (7.88). Instead of (7.92)let us consider

T J ( 12 x)J (− 1

2 x) ≈x2→0

∑n,k

Ckn(x

2)xµ1 . . . xµn Oµ1...µnk (7.99)

where the sum is taken over all twist two operators Ok and all even spins n. Using(7.93) and defining

2nqµ1 . . . qµn

(−q2)n+1Ck

n(q2) = i

∫d4x exp(iqx)xµ1 . . . xµn Ck

n(x2) (7.100)

the leading contribution to T (q2, P · q) in the Bjorken limit can be written as

T (q2, P · q) = 1

−q2

∑n,k

x−nBJ Ck

n(q2)Ak

n (7.101)

Note that Ckn(q

2) are defined so that they are dimensionless for twist two operatorsin the expansion of the product of two currents with dimension dJ = 3. Thereforein the case of canonical scaling Ck

n(q2) = Ck

n are constants. The coefficientsCk

n(q2) can be related to the moments of the structure function W (q2, P · q) if

we integrate T (q2, xBJ) over xBJ, for fixed q2. Remembering that it is an analyticfunction of xBJ in the complex xBJ-plane with a branch cut along the real axis for


Fig. 7.3.

|xBJ| < 1 we integrate over the contour shown in Fig. 7.3. Since the discontinuityof the T is just 1

2 W (q2, P · q) we have (x ≡ xBJ)

1

2π i

∮C

dx xm−1T (q2, x) = 1

4π

∫ 1

−1dx xm−1W (q2, x) (7.102)

and on the other hand

1

2π i

∮C

dx xm−1T (q2, x) = 1

−q2

∑k

Ckm(q

2)Akm (7.103)

Finally we get

Mm(q2) = (1/2π)

∫ 1

−1dx xmνW (q2, x) =

∑k

Ckm(q

2, µ2, g)Akm(µ

2, g) (7.104)

In (7.104) we have written down explicitly the dependence on the renormalizationpoint and the coupling constant.

Thus we get a relationship between moments of the deep inelastic structurefunction and the coefficient functions. Using the crossing properties of the structurefunction under q →−q or x →−x

W (q2, x) = −W (q2,−x) (7.105)

we can rewrite (7.104) for even m in terms of the integral over the physical regionin the electroproduction process, 0 < x < 1 (for odd m the integral (7.104)vanishes). We also note that (7.104) is meaningful for values of m such that theintegral converges. This may not be the case for low moments: for fixed q2 thelimit x → 0 corresponds to ν →∞, where ν = P · q ≈ s and s is the cms energysquared in the forward Compton amplitude. At high energy the imaginary part ofthis amplitude is expected to behave as sα, α ≈ 1, so that W (q2, x) ∼ x−1.


An important virtue of the OPE is that it allows us to factorize out the q2

dependence in quantities like Mm which depend on q2 and P2 = m2. Thisdependence can then be studied by means of the RGE. Similar analysis is notpossible directly for Mn(q2, P2 = m2) because the scale transformation q → ρq,P → ρP relates Mns for different q2s and m2s.

As we know, exact Bjorken scaling gives

Ckn(q

2) = const. (7.106)

Interactions can modify this behaviour. In the next section we shall study the q2

dependence of the Wilson coefficients using RGE.

7.6 Renormalization group and OPE

Renormalization of local composite operators

We are used by now to working with Green’s functions involving not onlyelementary fields but also composite operators, for example,

G(n)O (x1, . . . , xn, x) = 〈0|T�(x1) . . . �(xn)O(x)|0〉 (7.107)

or in momentum space

(2π)4δ(p1 + · · · + pn + p)G(n)O (p1, . . . , pn, p)

=∫

d4x exp(ipx)n∏

i=1

d4xi exp(ipi xi )G(n)O (x1, . . . , xn, x) (7.108)

The generating functional formalism and perturbation theory rules can be extendedto such cases if we introduce sources (x) coupled to operators O(x). Thefunctional

W [J, ] =∫

D� exp

[iS + i

∫d4x (J�+ O)

](7.109)

generates these new Green’s functions. Connected Green’s functions are obtainedfrom the functional Z and the Legendre transformation performed on sources J(only) generates 1PI Green’s functions with O insertions. As usual one has to takederivatives with respect to and then set = 0. Feynman rules for verticesinvolving elementary fields and the composite operators O(x) can be derived inexactly the same way as in Chapter 2 or Chapter 3. The new Green’s functionscalculated perturbatively, in general, require renormalization. The countertermspresent in the lagrangian for the renormalization of the Green’s functions involvingonly elementary fields are not sufficient for eliminating divergences in the Green’s

7.6 Renormalization group and OPE 257

functions with composite operator insertions. Additional operator renormalizationconstants are needed

OB(x) = ZOOR(x) (7.110)

If we want to express OR in terms of the renormalized fields the wave-functionrenormalization constants will explicitly appear, for example,

(�2B)B = Z�2(�2

B)R = Z�2 Z3(�2R)R (7.111)

Only in exceptional cases like conserved currents or currents of softly brokensymmetries does ZO = 1 (see Section 10.1). Relations (7.110) and (7.111)correspond to a replacement

(x)OB(�iB(x)) = OR(�i

R)+ (ZO Zi/23 − 1)OR(�i

R) (7.112)

in the generating functional, (7.109). We have taken O(x) to be of the i th orderin the elementary fields �(x). The last term in (7.112) is the new counterterm.The ZO can be calculated in perturbation theory by calculating the divergentGreen’s functions with operator insertions. The superficial degree of divergence(see Section 4.2) DO of �(n)

O differs from D of �(n) and reads

DO = D + (dcanO − 4) (7.113)

where dcanO is the canonical dimension of O. As an example let us consider λ�4

theory and discuss a single insertion of O(x) = 12�

2(x) (Itzykson & Zuber 1980)or an insertion of O(x) = �(x)�(x) in QCD. The divergent Green’s functionswith this insertion are �

(2)�2 and �

(2)��

with DO = 0. It is an easy exercise to calculateZO in these simple cases in the lowest order of perturbation theory. The relevantFeynman diagrams are shown in Fig. 7.4, where the vertex ⊗ is in both cases justone and for simplicity we can take the momenta of the external legs to be equal.

Of course, new renormalization constants require new renormalization condi-tions. For instance, we may impose

�(2)�2 (0, 0) = 1 and �

(2)��

(0, 0) = 1 (7.114)

in agreement with the lowest order.If an operator O is multiplicatively renormalized as in (7.110) then the 1PI n-

point Green’s functions with a single O insertion satisfy

�(n)O,R(p1, . . . , pn, p, λ, µ,m) = Z−1

O Zn/23 �

(n)O,B(p1, . . . , pn, p, λB,mB) (7.115)

Simple multiplicative renormalization, (7.110), is actually not always sufficient.For instance, it may not eliminate divergences from some Green’s functionsinvolving more then one insertion of O. A well-known example is the vacuum


Fig. 7.4.

expectation value of two electromagnetic currents Jµ(x) which requires a subtrac-tion although Z J = 1 (see Section 10.1).

Another complication arises for the following reason. Using the arguments ofSection 4.2 concerning the necessary counterterms in the standard renormalizationprogramme we can convince ourselves that an operator in the lagrangian withdimension d requires, in general, all possible counterterms with dimensions ≤dallowed by the symmetries of the bare lagrangian. If we limit our discussionto Green’s functions involving only one insertion of an operator O the same istrue for the renormalization of such composite operators. Therefore if there existseveral operators with the same quantum numbers as the operator O and with theircanonical dimensions lower or equal to the dimension of the operator O, they willmix with O under the renormalization. Thus, (7.110) takes a matrix form

OBn = (ZO)nkOk (7.116)

and correspondingly

�(n)On,R

(p1, . . . , pn, p, λ, µ,m) = (Z−1O )nk Zn/2

3 �(n)Ok,B

(p1, . . . , pn, p, λB,mB)

(7.117)Note that the matrix Zi j is not symmetric: Zi j = 0 if dim Oi < dim O j .

From (7.117) we can derive for Green’s functions with an operator insertion theRGE analogous to (6.9) and (6.24), for instance,{[

− ∂

∂t+ β(λ)

∂

∂λ+ γm

∂

∂m− nγ + D

]δi j + γ

i jO

}�

(n)O j

(et pi , λ(µ), µ,m) = 0

(7.118)where

γi jO = −

(µ

d

dµ(ZO)

ik

)(Z−1

O )k j


is the anomalous dimension matrix for the operators Ok and D is the dimension ofthe Green’s function (equal to the canonical dimension of all operators involved)and the mass dependence has been neglected.

At this point it is worthwhile to observe that in perturbation theory there is nofundamental difference between the renormalization of the original lagrangian,(4.1), and of the composite operators. Both can be formulated in the samelanguage. For instance, the renormalization constants Z0 and Z1, defined inChapter 4 as renormalization of the lagrangian parameters mB and λB, could aswell be defined as the renormalization constants for the operators �2 and �4.According to (2.89), the n-particle Green’s function is, in fact, a superpositionof Green’s functions with the vertex and mass operator insertions (we can treat themass as one of the parameters in perturbation theory). The derivatives m2∂/∂m2

and λ∂/∂λ count then the number of such insertions (see also (7.47)) and we arriveto the same RG equations (6.9) and (6.24), with, for example, γm interpreted nowas the anomalous dimension of the operator �2 (7.49).

In the opposite direction, the renormalization of local composite operatorsdiscussed here can be phrased in the language of Chapter 4. We can interpretthe operators On as a perturbation of the lagrangian (4.1):

LB�4 → LB

�4 +∑

n

∫d4x CB

n OBn

and we can renormalize the new coupling constants: CBn = Zn

OCn . We then derivethe RGE for the �

(n)On

by a complete analogy with the derivation of (6.9) and (6.24)and, putting Cn = 1, we get, for example, (7.118).

RGE for Wilson coefficients

We now turn back to the OPE and study the q2 dependence of the Wilson coefficientfunctions by means of the RGE. Let us recall that in general

A( 12 x)B(− 1

2 x) ≈x2→0

∑i

C ABi (x, g(µ), µ)Oi (0) (7.119)

and therefore for any Green’s function with insertion of the product AB we have

�(n)AB(q, p1, . . . , pn, g(µ), µ) =

∑i

C ABi (q, µ)�

(n)Oi

(0, p1, . . . , pn, g(µ), µ)

(7.120)By applying the differential operator

D = − ∂

∂t+ β(g)

∂

∂g


to (7.120) with all momenta rescaled by exp(t) and using (7.118) we get{[− ∂

∂t+ β(g)

∂

∂g+ DA + DB − DO + γA + γB

]δi j − γ

i jO

}C AB

i (etq, g(µ), µ)

= 0 (7.121)

where DA, DB and DO are canonical dimensions of A, B and O respectively andγA, γB and γ

i jO are their anomalous dimensions. We have assumed here that oper-

ators A and B do not mix with other operators when undergoing renormalization.This is in particular trivially true for the electromagnetic currents. The solution tothe RGE for the Wilson coefficient is analogous to (6.26). Let us write it downfor some specific cases. Firstly take Ci dimensionless, i.e. DA + DB − DO = 0.In addition put γA = γB = 0 as for conserved currents. For the no-mixing caseγ

i jO = δi jγi we then have

Ci (etq, g(µ), µ) = exp[−γi (g+)t] exp

{−∫ g(t)

gdx [γi (x)− γi (g+)]/β(x)

}× Ci (q, g(t), µ) (7.122)

To proceed further one has to calculate Ci (q, g(t), µ), γi (g) and β(g) in per-turbation theory. In asymptotically free theories the UV fixed point g+ = 0,β(x) = (2/16π2)b1x3 + · · · and γi (x) = (4/16π2)γ 0

i x2 + · · ·. Putting q2 = µ2,t = 1

2 ln(q2/µ2) and expanding

Ci (µ, g(t), µ) = Ci (µ, 0, µ)[1 + C1i g2(t)/4π + · · ·]

in the leading order in g2 we get

Ci (q, g(µ), µ) = Ci (µ, 0, µ)[g2(t)/g2(µ2)]−γ 0i /b1 (7.123)

Thus for the moments of the structure function we have (from (7.104) when k = 1,i.e. if there is only one twist two operator in expansion (7.99))

Mi (q2)

Mi (q20 )=[

g2(q2)

g2(q20 )

]−γ 0i /b1

=[

ln(q2/�2)

ln(q20/�

2)

]+γ 0i /b1

(7.124)

In the last step we have used (7.53). Eq. (7.124) is valid for both q20 and q2 large as

compared to �2. For a non-zero UV fixed point g+ �= 0 (7.122) gives a power-likeviolation of scaling

Ci (q, g(µ), µ) ≈ (q2/µ2)−γi (g+)Ci (µ, g+, µ) (7.125)


In the case of operator mixing the solution to the RGE can be written in the matrixform (take g+ = 0)

C(q, g(µ), µ) ={

Tg exp

[ ∫ g(µ2)

g(q2)

dx γ (x)/β(x)

]}C(µ, g(q2), µ) (7.126)

where

C =

C1...

Cn

, γ = [γi j ]

and the Tg ordering, necessary since [γ (x1), γ (x2)] �= 0, is defined as follows

Tg exp

[ ∫ g

gdx

γ (x)

β(x)

]= 1 +

∫ g

gdx

γ (x)

β(x)

+ 1

2!

∫ g

gdx∫ x

gdy

γ (x)

β(x)

γ (y)

β(y)+ · · · (7.127)

OPE beyond perturbation theory

Strictly speaking, the use of the OPE to study the moments of the structure functionin deep inelastic lepton–hadron scattering, which is discussed in this section,goes beyond perturbation theory. This is because in the realistic case, when thestrong interactions are described by QCD, the matrix elements of the operatorstaken between hadronic states contain non-perturbative effects responsible for thebinding of quarks and gluons into hadrons. Nevertheless we assume the validityof the OPE and, moreover, that the coefficient functions C AB

n are calculable inQCD perturbation theory. This approach is implicitly based on the parton modelassumption discussed in Section 8.4. It is also in agreement with the possibility,very suggestive as the physical sense of the OPE formalism, that the contributionfrom short distances can be separated from that of large distances. One can thenexpect that to a good approximation the coefficient functions are calculable inperturbation theory while the non-perturbative effects are accounted for by thematrix elements of the operators On .

The same idea underlies the use of the OPE to account for the non-perturbativestructure of the QCD vacuum, pioneered by Shifman, Vainshtein & Zakharov(1979) and extensively discussed in the recent years. Non-perturbative effectsmanifest themselves through non-trivial vacuum expectation values of variousoperators which vanish in perturbation theory

〈0| :On: |0〉 = 0 if On �= I


The best-known examples are quark and gluon condensates

〈0| :��: |0〉 and 〈0| :GaµνGµν

a : |0〉

the first one is responsible for the spontaneous breaking of chiral symmetry in QCD(Chapter 9). This use of the OPE has been successful phenomenologically but itstheoretical status and possible limitations are still subject to some discussion (see,for instance, Novikov et al. (1984) and references therein).

The coefficient functions C ABn (q) can be found by calculating the appropriate

Feynman diagrams (Problem 7.5). Important for this programme is the theoremthat the coefficient functions in the OPE manifestly reflect the symmetries of thelagrangian whether or not they are symmetries of the vacuum (Bernard et al. 1975).

7.7 OPE and effective field theories

OPE is a convenient tool for discussing effective field theories. Let us considera renormalizable action containing some light field(s) ϕ with mass m and someheavy field(s) � with mass M (such that m � M)

S(ϕ,�) =∫

dx[Lϕ(x)+ L�(x)+ Lϕ�(x)

](7.128)

We are interested in calculating amplitudes involving only the light particles ϕ

with external momenta much smaller than M . The Green’s functions for processesinvolving only light particles are generated by the functional (recall (2.84))

W [J ] = N∫

DϕD� exp

{iS(ϕ,�)+ i

∫dx J (x)ϕ(x)

}= N

∫DϕD� exp

{i∫

dx[Lϕ(x)+ J (x)ϕ(x)

]}×

∞∑k=0

ik

k!

∫dy1 . . . dyk Lϕ�(y1) . . .Lϕ�(yk) exp

{i∫

dx L�(x)

}.

(7.129)

From now on, we assume for simplicity that the interaction lagrangian Lϕ� islinear in �, i.e. Lϕ� = L(ϕ)�. (The derivatives acting on � can be removed

7.7 OPEs and effective field theories 263

by integration by parts.) In such a case, we can write

W [J ] = N∫

Dϕ exp

{i∫


]}×

∞∑k=0

ik

k!

∫dy1 . . . dyk L(y1) . . . L(yk)

×∫

D��(y1) . . . �(yk) exp

{i∫

dx L�(x)

}(7.130)

= N ′∫

Dϕ exp

{i∫


]}×

∞∑k=0

ik

k!

∫dy1 . . . dykL(y1) . . . L(yk)G

�(y1, . . . , yk)

Comparison with (2.130), (2.131) and (2.137) allows us to ‘exponentiate’ withsimultaneous replacement of ordinary Green’s functions by the connected ones

W [J ] = N ′∫

Dϕ exp

{i∫


]+

∞∑k=0

ik

k!

∫dy1 . . . dyk L(y1) . . . L(yk)G

�conn(y1, . . . , yk)

}(7.130a)

We have thus arrived at the effective theory described by a non-local action. As faras the processes involving only external ϕ particles are concerned, it is perfectlyequivalent to the the underlying theory (7.128). The important point is that theeffective region of integration over y involves distances much shorter than theregion of integration over x .

Now, the OPE has to be performed. The OPE hypothesis (7.54) can beformulated in the path integral language as follows. Let A(x) and B(x) be somemonomials in the fields ϕ(x) and their derivatives, belonging to a complete basis ofsuch monomials {Ok(x)}. Then there exists a set of Wilson coefficients {Ck

AB(x)}such that†∫

DϕA(x)B(y) exp

{iS[ϕ] + i

∫dx J (x)ϕ(x)

}=∑

k

CkAB(x − y)

∫Dϕ Ok

(x + y

2

)exp

{iS(ϕ)+ i

∫dx J (x)ϕ(x)

}(7.131)

† Strictly speaking, it corresponds only to OPE for T -products. All the local field products here are tacitlyassumed to involve counterterms which in the operator language would correspond to their normal ordering(see (2.105)–(2.107)). Moreover, if A or B involve derivatives, the series in the r.h.s. of (7.131) is not in aone-to-one correspondence with that of (7.54), which is due to the features described below (2.56).


One can prove the validity of a similar expansion for multiple products likeA(x)B(y)C(z) by appropriate differentiations of (7.131) with respect to J (z)and z (according to the structure of C(z)), and applying (7.131) to each termon the r.h.s. In a similar way, one can show that if A(x)B(y) correspondsto a series

∑Ck

AB Ok and C(x)D(y) corresponds to a series∑

CkC D Ok , then

A(x)B(y)C(z)D(w) corresponds to the product[∑k

CkAB(x − y)Ok

(x + y

2

)][∑n

CnC D(z − w)On

(z + w

2

)](7.132)

Finally, one arrives at the conclusion that it is possible to replace the productsL(y1) . . . L(yn) in (7.130a) by sums of the form

L(y1) . . . L(yk) −→∑

l

C (k)l (y1, . . . , yk)Ol(ymean) (7.133)

in spite of the fact that they stand under the exponent. This latter conclusion wouldbe rather obvious in the operator language. In the next step, we integrate thecoefficients C (k)

l with the connected Green’s functions keeping all the ymeans fixed,i.e. performing only k−1 integrations for each k-point function. The results of thisintegration are independent of ymean, which follows from translational invariance.In this way we obtain

W [J ] = N ′∫

Dϕ exp

{iSeff + i

∫dx [ Lϕ(x)+ J (x)ϕ(x)]

}(7.134)

with Seff given by

Seff =∫

dx

[Lϕ(x)+

∑l

1

Mn(l)Cl Ol(x)

], n(l) ≥ 1. (7.135)

and

Cl = Mn(l)∑

k

ik

k!

∫dy1 . . . dyk C (k)

l (y1, . . . , yk)G�conn(y1, . . . , yk)δ(ymean)

(7.136)The factors Mn(l) have been extracted above just to keep Cl dimensionless. Thecoefficients Cl are customarily also called Wilson coefficients. However, oneshould stress the important difference with respect to the original (true) Wilsoncoefficients defined by (7.133): in (7.136) the latter are integrated with the Green’sfunctions for heavy particles. Therefore, the coefficients Cl are constants, whichcan be interpreted as additional independent coupling constants for the effectiveinteraction vertices Ol . Lϕ stands for terms which would have n(l) ≤ 0.

It remains to show that Lϕ (which may contain M independent terms,logarithms and positive powers of large mass M) can be absorbed into Lϕ by


renormalization of its fields and parameters, and that Cls are convergent (orbehaving at most logarithmically) in the limit M → ∞. This requires theintroduction of some restrictions on the action S in (7.128). We assume that thetheory described by Lϕ is

(i) renormalizable, and the symmetries defining† Lϕ are also the symmetries of Sincluding the possible gauge fixing terms for �, but not for ϕ (see Weinberg (1980));

(ii) asymptotically scale-invariant, up to logarithmic corrections.

By assumption (i) Lϕ contains only terms allowed by symmetries of S (weassume the absence of anomalies). This guarantees that Lϕ can be absorbed intoLϕ by appropriate redefinition of fields and parameters in Lϕ .

Determining the behaviour of the Cls in the limit M → ∞ is based on the factthat the (Euclidean) connected Green’s functions for the � particles exponentiallyvanish when splitting among their arguments becomes much larger than 1/M .Thus, the behaviour of the Cls at M →∞ is determined by the leading behaviourof (Euclidean versions of) C (k)

l (y1, . . . , yk) at points where all the yi almostcoincide. Assumption (ii) tells us that the latter behaviour is independent ofdimensionful parameters present in Lϕ , up to logarithmic corrections. Hence, theleading behaviour of the Cls at M →∞ has the same property.‡ This is equivalentto saying that they behave at most logarithmically, because powers of 1/M havebeen already extracted in (7.135) to make the Cls dimensionless.

An important point omitted in the above discussion is checking whether theintegrals present in (7.136) are not divergent in spite of their proper behaviourat M → ∞. In fact, they are UV divergent, but these divergences are expectedto cancel among different integrals when counterterms for the full theory definedby the lagrangian (7.128) are properly included. Checking this would requireexpressing C (k)

l in terms of the ϕ-particle Green’s functions (and their derivatives)with the help of (7.131). We are going to bypass this point here.

The action Seff in (7.135) is, in principle, non-renormalizable because it containsterms of arbitrary dimension. However, as we will see below, the renormalizationprogramme can be effectively carried out. Moreover, since the contributions to theamplitudes from higher-dimensional Ols are suppressed§ by powers of (externalmomenta)/M , only a finite number of Ols needs to be considered in practice. Theparameters (couplings) Cl undergo renormalization and also evolve according totheir RGEs which do not depend on the anomalous dimensions of the original

† A renormalizable theory can be defined by specifying its field content and its symmetries. The action is builtfrom all possible terms of dimension ≤ 4 that are invariant under the symmetries. Of course, not every set offields and symmetries can give us a renormalizable theory.

‡ Provided the interaction L�ϕ does not contain coupling constants growing with M . Such a situation occurswhen Higgs–fermion couplings in the Standard Model are considered in the process of heavy fermiondecoupling.

§ This can be easily shown on dimensional grounds.


operators in the OPE (7.133). They can be considered to be new free parametersto be determined from experiment or, if the full theory is known, they can berecovered by requiring the equality of amplitudes generated by (7.128) and (7.135).Only in this sense are they dependent on m/M and other parameters present in(7.128).

The reason for introducing the effective action (7.135) is the fact that theperturbative expansion of low energy amplitudes generated by (7.128) is in practicean expansion in powers of (coupling constants)· ln[(external momenta)/M] whichmay be too large to be expansion parameters. Replacing (7.128) by (7.135)combined with the RGE evolution allows us to extract large logarithms from allorders of the perturbation series. There are four main steps of this procedure:

First, one recovers the renormalization constants in Seff determined by therequirement of finiteness of amplitudes generated by Seff. The renormalizationconstants in Lϕ may be different than in the underlying theory (7.128). The sumover l in (7.135), when expressed in terms of renormalized quantities, takes theform ∑

l

1

Mn(l)

[∑l1

Zl1lCl1 +∑l1l2

Zl1l2lCl1Cl2 + · · ·]

Zl Ol (7.137)

where the Zls are products of renormalization constants of the fields presentin Ols, and Zl1...l j l are additional renormalization constants necessary to canceldivergences in diagrams with j Ol-vertices. All the Zs in the above expressiondepend on m/M and the couplings present in Lϕ , but not on the coefficientsCl . It is easy to convince oneself that Zl1...l j l is proportional to M−p, wherep = n(l1)+ · · · + n(l j )− n(l). It vanishes for p < 0 which can be justified† usingthe arguments of Section 4.2. Therefore, only a finite number of renormalizationconstants has to be found when one works up to a given order in 1/M . This iswhy the renormalization programme can, in practice, be carried on in the theorydescribed by (7.135).

Next, one finds the coefficients Cl (and, in principle, also the renormalizedparameters in Lϕ) by requiring equality of amplitudes generated by (7.128) and(7.135). The renormalization scale µ has then to be set equal to µ ∼ M , in orderto avoid large logarithms.‡

The third step is deriving and solving the RGEs for the effective action parame-ters, which results in expressing these parameters renormalized at the scale µF ∼† According to (4.42), the superficial degree of divergence of a diagram with Ol1 , . . . , Ol j -vertices is equal to

D = 4− B − 32 F + n(l1)+ · · · + n(l j ). After removing subdivergences, it may require counterterms which

are polynomials in external momenta only of order≤ D. Such counterterms are generated by Ol s involving Bbosonic fields, F fermionic fields and no more than D derivatives. Such Ol s have n(l) ≤ D+B+(3/2)F−4,and therefore n(l1)+ · · · + n(l j )− n(l) ≥ 0.

‡ We restrict ourselves to the µ-dependent renormalization schemes such as MS or the µ-subtraction scheme.


(external momenta) in terms of their counterparts renormalized at µI ∼ M . Thelatter have been already found by ‘matching’ in the previous step and serve as initialconditions for the RGEs. As long as one works in the framework of dimensionalregularization, the renormalized Cls should be rescaled by appropriate powers ofµε (as has been done with λR in Section 6.1) in order to keep them dimensionless.The RGEs are then derived, for example, from the equation d/dµ Seff = 0,and the methods† of Section 6.2 are used for determining the µ-derivatives ofrenormalization constants. For simplicity, let us restrict our attention to the leadingorder in 1/M . Then, only the first of the sums in the bracket of (7.137) needs to beconsidered and, interpreting Cls as new coupling constants, we get

µd

dµCl(µ) =

∑k

γkl(g(µ))Ck(µ), (7.138)

where the anomalous dimension matrix γ is determined by

γkl(g) = −∑

j

(µ

d

dµZkj

)(Z−1) jl . (7.139)

The last step is calculating the desired amplitudes with help of Seff renormalizedat µF ∼ (external momenta). This is done either perturbatively or with use of somenon-perturbative methods (Buchalla, Buras & Harlander 1990).

In principle, all possible terms allowed by symmetries of (7.128) are expected toappear in the expansion (7.135). Even at the leading order in 1/M , their numberis usually very large. Therefore, at the beginning, elements of all the four stepsare performed simultaneously in order to select the set of operators Ol that maycontribute to the desired amplitudes at the desired order (even by their influence onthe RGE evolution of some directly contributing Ol). One should also rememberthat even the coefficients Cl vanishing at µI can acquire non-zero values at µF, dueto the non-diagonal character of the mixing in (7.137).

As already mentioned, the outlined procedure allows us to extract large loga-rithms from all orders of the perturbation series. In the following, the meaning ofthis statement will be made more precise. Let us assume that Lϕ contains onlyone (dimensionless) coupling constant g. Then, it is convenient to solve the RGEsusing the variable v = g2 ln(µ2

I /µ2) instead of µ itself. The variable v is treated

as independent of g(µI), and all the RGEs are solved perturbatively in g(µI). Thecoupling constant g(µ) is therefore determined, up to a given order in g(µI), by

† It may not always be possible to define a mass-independent renormalization scheme in a theory describedby Seff. Then the methods of Section 6.2 have to be modified. Here we consider only those cases in whichit is possible to avoid the appearance of dimensionful parameters. Sometimes this requires changing thenormalization of some of the Ol s, for example, multiplying them by m.


the series

g(µ) = g(µI)

∞∑n=0

g2n(µI) fn(v), (7.140)

where fn(v) are certain functions found with help of the equation (see (6.6))

µd

dµg(µ) = β(g). (7.141)

The coupling g(µ) from (7.140) is then inserted into the RGE for the coefficientsCl , (7.138). Since µd/dµ = −2g2(µI)d/dv, knowledge of γ and β up to (n+1)thorder in g2 is necessary for determining the Cls up to nth order in g2(µI) andexactly in v. Thus, if

• the renormalization constants (first step) are found up to (n + 1)th order in g2,• the matching at µI (second step) is performed up to nth order in g2(µI),• the final amplitude (last step) is calculated up to nth order in g2(µF) and expanded with

help of (7.140),

we obtain the final answer up to nth order in g2(µI) and exactly in vF =2g2 ln(µI/µF). This is the precise meaning of the statement that all largelogarithms are extracted from the perturbation series. Such a result could neverbe obtained with help of perturbative calculus based on the action (7.128) itself.The price we have to pay is the necessity of truncating the series in 1/M , but thisis often a much better approximation than truncating the series in g(µI) which hasto be done anyway.

There are two main domains of applications of the described technique. Firstof them covers low energy electroweak processes involving hadrons (see, forexample, Grinstein, Springer & Wise (1990)). There, the role of heavy particles isplayed by the W± and Z0 bosons, the top quark and the Higgs bosons. The couplingwhich becomes too large after multiplication by the logarithm is the QCD gaugecoupling. In some processes, the b and c quarks are also treated as heavy. Thenthe decoupling is done successively and the matching between several ‘effectivetheories’ is performed (see, for example, Gilman & Wise (1979)). (More details ofthe effective theory approach are presented in Sections 12.6 and 12.7). The otherdomain covers phenomena taking place much above the electroweak scale (see, forexample, Wilczek & Zee (1979)). There, the coupling constants are smaller but theevolution is usually longer. In fact, even the coupling constant splitting after thespontaneous symmetry breakdown in GUTs can be most easily understood in termsof the effective theory arising after the heavy gauge boson decoupling (Weinberg1980). It is important to notice that after decoupling the resulting effective theoryusually has less symmetries than the original one (as in the examples mentionedabove).

Problems 269

Problems

7.1 (a) Check that the set of 15 infinitesimal transformations defined by (7.26) is closedunder commutation. Check the differential representation of the generators

i[Pµ, f (x)] = ∂µ f

i[Mµν, f (x)] = (xµ∂ν − xν∂µ) f

i[D, f (x)] = xµ∂µ f

i[Kµ, f (x)] = (x2gµν − 2xνxµ)∂ν f

where f (x) is a dimensionless scalar function and

f (x ′) = U f (x)U−1

where

U (a) = exp(iaµPµ), U (ε) = exp(iεD),

U (ω) = exp(− 12 iωµν Mµν), U (b) = exp(ibµKµ)

for x ′µ = xµ + εµ(x).(b) Check the algebra

[Mµν, D] = [Pµ, Pν] = [Kµ, Kν] = [D, D] = 0

[Pµ, D] = iPµ

[Kµ, D] = iKµ

[Mµν, Kλ] = i(gµλKν − gνλKµ)

[Kµ, Pν] = 2i(−gµν D + Mµν)

[Mµν, Pλ] = −i(gµλPν − gνλPµ)

[Mαβ, Mµν] = −i(gαµMβν − gβµMαν + gαν Mµβ − gβν Mµα)

(c) Check the infinitesimal transformation law for a field � under the specialconformal transformation

δµ�(x) = (x2∂µ − 2xµxν∂ν − 2xµd� − 2xν�µν)�(x)

7.2 Study conformal symmetry in two-dimensional quantum field theories (Belavin,Polyakov & Zamolodchikov 1984).

7.3 When the dilatation current is not conserved one may still define a time-dependentdilatation charge by

D(t) =∫

d3x xµ�0µ(x) = t P0 +

∫d3x xi�

0i (x)

Check that D(t) generates the same dilatation transformation (7.14) on the fields asin the scale-invariant theory (use the canonical formula (7.16) for Dµ(x)). Obtain the


commutators of D(t) with the Poincare generators

i[D(t), Pα] = Pα − gα0∫

d3x ∂µDµ(x)

i[D(t), Mαβ ] =∫

d3x (gα0xβ − gβ0xα)∂µDµ(x)

7.4 (a) Using the Wick expansion and then the Taylor expansion of the well-definednormal-ordered operator products, prove (7.59) in the free scalar field theory.(b) In the free quark model show that

T J aµ(x)J b

ν (0) ∼xµ→0

3δab(gµνx2 − 2xµxν)

(2π)4(x2 − iε)4I + dabc

2π2

εµναβxα Aβc (0)

(x2 − iε)2

+ cabc

2π2

xρ Sµρνσ Jσc (0)

(x2 − iε)2+ · · ·

T J aµ(x)Ab

ν(0) ∼xµ→0

dabc

2π2

εµναβxα Jβc (0)

(x2 − iε)2+ cabc

2π2

xρ Sµρνσ Aσc (0)

(x2 − iε)2+ · · ·

where

J aµ(x) = :�c(x)γµ

12λa�c(x):

Aaµ(x) = :�c(x)γµγ5

12λa�c(x):

the repeated index c implies a sum over colours; λa are SU (3) flavour matrices

14λaλb = 1

2 (dabc + icabc)λc

γµγργν = (Sµνρσ + iεµρνσ γ5)γσ

Sµρνσ = (gµρgνσ + gρνgµσ − gµνgρσ )

Also derive the light-cone expansion for commutators of these currents (for example,Ellis (1977)).

7.5 Obtain the expansion

i∫

d4x exp(iqx)T jµ(x) jν(0) = (qµqν − q2gµν)

(− 1

4π2ln−q2

µ2+ 2m�

q4�� + · · ·

)

where jµ = �γµ�. The first term is given by the diagram

〈0| jµ jν |0〉 = = C (0)I 〈0|I |0〉

Problems 271

the second by

〈�| jµ jν |�〉 = = C (0)��

〈�|��|�〉

= C (0)��

· = C (0)��

Observe that to this order in α taking the appropriate matrix element singles out oneparticular contribution to the OPE. Extend the calculation to first order in α; in additionto C (1)

I and C (1)��

given by the same matrix elements (now taken to the first order in α)calculate also the two-gluon matrix element

〈g| jµ jν |g〉 = C��〈g|��|g〉 + CG〈g|GαµνGµν

α |g〉

where C (0)��

, is given by the zeroth order calculation, and find C (1)G .

8

Quantum chromodynamics

8.1 General introduction

Renormalization and BRS invariance; counterterms

QCD is a theory of interactions of quarks and gluons. Its lagrangian density hasbeen discussed in Sections 1.3 and 3.2. In terms of bare quantities it reads

L = − 14 Gα

µνGαµν +∑

f

� f (iD/ − m f )� f

+ gauge-fixing term + Faddeev–Popov term (8.1)

where the sum is over all quark flavours and the last two terms are given in (3.46)for the class of covariant gauges. The quarks are assumed to transform accordingto the fundamental representation: each flavour of quark is a triplet of the colourgroup SU (3). Gauge bosons transform according to the adjoint representation, sothat there are eight gluons.

In this section we discuss the theory formulation in covariant gauges. The useof the axial gauge n · Aα = 0, n2 < 0, or light-like gauge, n2 = 0, where n isin each case some fixed four-vector, is also often very convenient in perturbativeQCD calculations. QCD formulation in the axial gauge n2 < 0 can be given ina similarly precise form to that in covariant gauges (for example, Bassetto et al.(1985)). In light-like gauges the general proof of renormalizability is still lackingbut order-by-order calculations have been consistently performed.

Let us concentrate on the renormalization programme. As in QED, the crucialissue is the gauge invariance or strictly speaking BRS invariance of the theory. Itcan be proved (for example, Collins (1984)) that the lagrangian is BRS-invariantafter renormalization. Let us therefore rewrite lagrangian (8.1) in terms of therenormalized quantities assuming the most general form consistent with BRS

272

8.1 General introduction 273

invariance.† We get (for one flavour)

L = − 14 Z3[∂µ Aα

ν − ∂ν Aαµ − g(Z1YM/Z3)c

αβγ Aβµ Aγ

ν ]2

+ Z2�iγµ[∂µ + ig(Z1/Z2)AαµT α]� − m Z0��

+ Z2η∗α[δαβ∂2 + ˜g(Z1/Z2)c

αβγ Aγµ∂

µ]ηβ − (1/2a)(∂µ Aβµ)2 (8.2)

where

gZ1YM/Z3 = gZ1/Z2 = ˜gZ1/Z2 (8.3)

If we use a renormalization scheme such as the minimal subtraction scheme inwhich the renormalized couplings satisfy

g = g = ˜g

then (8.3) implies the corresponding equality for the ratios of the renormalizationconstants. As in QED there is no need for a gauge-fixing counterterm.

Relation (8.3) follows from the requirement of the BRS invariance for thelagrangian (8.2). Formally it can be derived from the Slavnov–Taylor identities(Ward identities for a non-abelian gauge theory) which themselves follow from theBRS invariance.

The correspondence between the renormalized quantities and the bare quantitiesof lagrangian (8.1) is the following

AαBµ = Z1/2

3 Aαµ �B = Z1/2

2 � aB = Z3a

ηB = Z 1/22 η η∗B = Z 1/2

2 η∗ mB = m Z0/Z2

}(8.4)

gB = gZ1YM/Z3/23 = gZ1/Z2 Z1/2

3 = ˜gZ1/Z2 Z1/23 (8.5)

Lagrangian (8.1) is invariant under the BRS transformation (3.76)

δBRS�B = −igBT α�BηαB�B ≡ �B(s�B)

δBRS�B = igB�BT α�BηαB ≡ �B(s�B)

δBRS AαBµ = �B(∂µη

αB + gBcαβγ η

β

B Aγ

Bµ) ≡ �B(s AαBµ)

δBRSηαB = 1

2 gBcαβγ ηβ

Bηγ

B�B ≡ �B(sηαB)

δBRSη∗αB = −(1/a)�B∂µ Aµα

B ≡ �B(sη∗αB )

(8.6)

† See also the beginning of Chapter 5.

274 8 Quantum chromodynamics

and lagrangian (8.2) under

δBRS� = −igZ1T α�ηα�

δBRS� = igZ1�T α�ηα

δBRS Aαµ = �(∂µη

α Z2 + gZ1cαβγ ηβ Aγµ)

δBRSηα = 1

2 gZ1cαβγ ηβηγ�

δBRSη∗α = −(1/a)�∂µ Aµα

(8.7)

Invariance of the lagrangian (8.2) under (8.7) can be checked explicitly but it alsofollows immediately from the invariance of bare lagrangian (8.1) under (8.6) ifrelations (8.3), (8.4) and (8.5) are inserted into (8.6) and if � is identified as

� = (1/Z1/23 Z1/2

2 )�B

We note the presence of the divergent renormalization constants Zi in the BRStransformation (8.7). This is actually what is necessary for finiteness of thecomposite operators δ�, δ�, δAα

µ and δηα (Collins 1984).

Lagrangian (8.2) can be rewritten introducing counterterms (we take g = g = ˜g)

L = − 14 Gα

µνGµνα + �iD/� − m�� − (1/2a)(∂µ Aµ

β )2

+ η∗α(δαβ∂2 + gcαβγ Aµγ ∂µ)η

β

+ (Z2 − 1)�i/∂� − (Z1 − 1)g�γ µ AαµT α� − m(Z0 − 1)��

+ (Z2 − 1)η∗α∂αβ∂2ηβ + (Z1 − 1)gη∗αcαβγ Aµγ ∂µη

β

− 14(Z3 − 1)(∂µ Aα

ν − ∂ν Aαµ)

2 + 12(Z1YM − 1)g(∂µ Aα

ν − ∂ν Aαµ)c

αβγ Aµβ Aν

γ

− 14 g2(Z2

1YM Z−13 − 1)cαβγ Aβ

µ Aγν cαρσ Aρµ Aσν (8.8)

The Feynman rules for the vertices of (8.8) have been derived in Section 3.2. Theyare given in Appendix B. The complete set which includes the counterterms isgiven in Appendix C (in the notation of Chapter 12).

Asymptotic freedom of QCD

The RGE, the notion of the effective coupling constant and its role in determiningthe behaviour of a quantum field theory at different momentum scales have beenextensively discussed in Chapter 6. In this context the behaviour of observablecross sections has been discussed in Chapter 7 in the framework of the OPE whichprovides a systematic way of factorizing effects at different mass scales.

QCD is an asymptotically free theory: the coupling constant decreases as thescale at which it is defined is increased. Thus, the behaviour of the Green’s


functions when all the momenta are scaled up with a common factor is governedby a theory where g → 0. This can explain the success of the parton model indescribing the scaling phenomena in the deep inelastic processes. Only theorieswith a non-abelian gauge symmetry can be asymptotically free in four space-timedimensions (Coleman & Gross 1973).

The β-function in QCD defined as

β(α) = µ(d/dµ)α(µ2)|αB fixed, α = g2/4π

is given by expansion (6.45)

β(α) = α

[α

πb1 +

(α

π

)2

b2 +(α

π

)3

b3 + · · ·]

(6.45)

and the coefficients bi are calculable perturbatively. Using dimensional regulariza-tion and a mass-independent renormalization scheme we have (6.33) and (6.35):β(α) is totally determined by the residua of the simple UV pole of Zα (beyond theone-loop order we need to calculate a non-leading UV divergence) where accordingto (8.5) one can use one of the relations

Zα = Z21YM/Z3

3 = Z21/Z2

2 Z3 = Z 21/Z 2

2 Z3 (8.9)

Diagrams contributing to different renormalization constants are determined by thestructure of counterterms in lagrangian (8.8). For instance at the one-loop level wehave the contributions shown in Fig. 8.1 (sums over flavours and over differentpermutations are omitted). Tadpole diagrams have been ignored because they donot contribute when dimensional regularization is used. One can use the mostconvenient way of calculating Zα.

As compared to QED (Section 5.2) the new features in QCD Feynman diagramcomputation are the group theory factors present for each diagram. The basic ingre-dients are SU (3) generators in the fundamental three-dimensional representationconventionally taken as

(T a)bc = 12(λ

a)bc

where λs are the Gell-Mann matrices, Tr[λaλb] = 2δab and in the regular (adjoint)eight-dimensional representation

(T a)bc = −icabc

where cabcs are the SU (3) structure constants. Both appear in the QCD vertices,and in the diagram calculation one often encounters the following combinations:∑

a,c

(T aR )bc(T

aR )cd = (T 2

R )bd = CRδbd (8.10)


Fig. 8.1.

and

∑c,d

(T aR )dc(T

bR )cd = Tr[T a

R T bR ] = TRδ

ab (8.11)

CR is the eigenvalue of the quadratic Casimir operator in the representation R and


for SU (N )

CF = (N 2 − 1)/2N F ≡ fundamentalCA = N A ≡ adjoint

The trace TR is

TF = 12

TA = N

So far the first three terms in the expansion (6.45) have been calculated and in theminimal subtraction scheme they are (for SU (N ) and n flavours)†

−b1 = 116 CA − 2

3 TFn

−b2 = 1712C2

A − 12CFTFn − 5

6CATFn

−b3 = 28571728C3

A − 1415864 C2

ATFn + 79432CAT 2

F n2 − 205288CACFTFn

+ 44288CFT 2

F n2 + 116C2

FTFn

(8.12)

The QCD effective coupling constant is given by expression (6.51) in the leadinglogarithm approximation, expression (6.53) in the next-to-leading approximation,and generally by (6.52). It is often convenient to express the result of a QCDcalculation in terms of the dimensional parameter � instead of the dimensionlessα. This has been discussed in Section 7.3.

The Slavnov–Taylor identities

As with the Ward identities in QED, their analogue in a non-abelian gauge theory– the Slavnov–Taylor identities for bare regularized or for renormalized Green’sfunctions – can be derived in several different ways. One can proceed as inSection 5.1: apply gauge transformation to the generating functional for full orconnected Green’s functions which is written as a functional of the physical fieldsources only. Then one derives the non-abelian analogue of (5.11) (Abers & Lee1973, Marciano & Pagels 1978). This form generates all the identities for theconnected Green’s functions but, because of the presence of ghost fields it is verycomplicated to obtain the analogous relation for the 1PI generating functional�(A, �, �). A more elegant method is based on the BRS transformation (Lee1976). In addition to sources J, α, α for the fields A, � and �, respectively, oneintroduces sources cα and c∗α for the ghost field η∗α and ηα as well as sources K ,L , L and M for the composite operators s A, s�, s� and sη that appear in the BRS

† The coefficient b1 was first calculated by Politzer (1973) and Gross & Wilczek (1973), the coefficient b2 byCaswell (1974) and Jones (1974) and the coefficient b3 by Tarasov, Vladimirov & Zharkov (1980).


transformation (8.6). This is to obtain identities that are linear in derivatives withrespect to the sources. Next one defines the generating functional

W [J, α, α, c, c∗, K , L , L, M]

=∫

D(A��ηη∗) exp

[iSeff + i

∫d4x (J A + α� + �α

+ c∗η + η∗c + K s A + s�L + Ls� + Msη)

](8.13)

Changing variables in the functional integral according to the BRS transformationand using the invariance

δSeff/δ� = 0

as well as the nilpotent relations

s2 A = s2η = s2� = s2� = 0

one obtains the desired general relation generating the Slavnov–Taylor identities.Since the ghost fields are treated on equal footing with the physical fields it is nowstraightforward to obtain similar relation-generating identities for the 1PI Green’sfunctions. As usual this is achieved by means of the Legendre transform

�[A, �, �, η, η∗, K , L , L, M] = Z [J, α, α, c, c∗, K , L , L, M]

−∫

d4x (J A + α� + �α + c∗η + η∗c)

(8.14)

where ηα = δZ/δc∗α, c∗α = −δ�/δηα etc. One can convince oneself that in thepresence of additional sources K , L , L and M the functional � still generates 1PIGreen’s functions.

The general relations whose derivation has just been briefly described containall the information about BRS invariance and are useful in formal manipulations.The simplest way, however, to get specific identities for the Green’s functions ofinterest is based on the observation that since BRS invariance is an exact symmetryof the theory all the Green’s functions are BRS invariant (for example, LlewellynSmith (1980)). For instance, let us consider the regularized Green’s function〈0|T Aα

µ(x)η∗β(y)|0〉. Using (8.6) we have

δBRS〈0|T Aαµ(x)η

∗β(y)|0〉 = 〈0|T (∂µηα(x)+ gcαβγ ηβ(x)Aγ

µ(x))η∗β(y)|0〉

− (1/a)〈0|T Aαµ(x)∂ν Aνβ(y)|0〉 = 0 (8.15)

8.2 The background field method 279

Differentiating (8.15) with respect to xµ and using the equation of motion for theGreen’s function 〈0|Tηα(x)η∗β(y)|0〉 which can be derived analogously to (2.56a)we conclude that

(1/a)∂µ

(x)∂ν(y)〈0|T Aα

µ(x)Aβν (y)|0〉 = −iδ(x − y)δαβ (8.16)

Result (8.16) implies that as in QED (see (5.14)) the longitudinal part of thepropagator is unchanged by the higher order corrections.

8.2 The background field method

This method is based on the idea of quantizing field theory in the presence of abackground field chosen so that certain symmetries of the generating functionalsare restored. Let us consider gauge theories. We start with the gauge-fixedfunctional integral (the dependence on other fields and on the ghost sources is notindicated explicitly)

W [J ] =∫

DA′µ exp(iSeff + iJ · A′), J · A′ =

∫d4x J α

µ A′µα (8.17)

and define, as usual,

Z [J ] = −i ln W [J ] (8.18)

and the effective action

F[ A] = Z [J ] − J · A, Aµα = δZ/δ J α

µ (8.19)

Manifest gauge invariance is lost because of the gauge-fixing procedure. However,we can restore it by introducing a background field with properly chosen transfor-mation properties. Let us consider first the gauge-invariant part of the action andintroduce the background field by the replacement

A′µ = Aµ + Aµ (8.20)

where Aαµ is the background field and Aα

µ is the quantum field (we recall thatAµ(x) = ig Aα

µ(x)Tα,�(x) = i�α(x)T α and we define Aµ(x) = igAα

µ(x)Tα).

Invariance of the action under the transformation (1.113)

δA′µ = Dµ�(x) = ∂µ�(x)+ [A′

µ(x),�(x)] (8.21)

implies invariance under two kinds of transformations on Aµ and Aµ which givethe same δA′

µ

quantum

δAµ = 0δAµ = Dµ�(x)+ [Aµ,�(x)], Dµ = ∂µ + [Aµ, . . .]

}(8.22)


background

δAµ = Dµ�(x), Dµ = ∂µ + [Aµ, . . .]

δAµ = [Aµ(x),�(x)]

}(8.23)

We can maintain the manifest gauge invariance of Seff with respect to thebackground transformations if the gauge-fixing function Fα[A,A] transformscovariantly under these transformations. For instance, choose

(Fα(Aµ,Aµ))2 = −(1/2a)(Dµ Aµ)2

α = −(1/2a)(∂µ Aαµ − gcαβγ Aµ

β Aγµ)

2 (8.24)

The Faddeev–Popov ghost term then takes account of

det Mαβ = det(δ Fα/δ�β)

where δ Fα corresponds to the quantum gauge transformation

δAαµ = (1/g)∂µ�

α + cαβγ�β(Aγµ + Aγ

µ)

and it also transforms covariantly under the background gauge transformation.The generating functional

W [J,Aµ] =∫

DAµ exp{iSeff[Aµ + Aµ] + iJ · A} (8.25)

is manifestly invariant under the background gauge transformations if we completethem with the transformation

δ J αµ = cαβγ�β J γ

µ (8.26)

The same is true for

Z [J,Aµ] = −i ln W [J,Aµ] (8.27)

and for

�[ Aµ,Aµ] = Z [J,Aµ] − J · A (8.28)

Aαµ = δ Z [J,Aµ]/δ J a

µ

if

δ Aαµ = cαβγ�β Aγ

µ

The next step is to find the relationship between the original �[ A] and the�[ Aµ,Aµ] in the presence of the background field. Changing the integrationvariables in (8.25) for W [J,Aµ]: Aµ → Aµ + Aµ one gets

W [J,Aµ] = W [J ] exp(−iJ · Aµ) (8.29)

8.2 The background field method 281

and correspondingly

Z [J,Aµ] = Z [J ] − J · Aµ (8.30)

Therefore

�[ Aµ,Aµ] = Z [J ] − J · Aµ − J · Aµ = �[Aµ + Aµ] (8.31)

and we conclude that the effective action �[ Aµ,Aµ] is the usual effective action�[ Aµ] evaluated at Aµ = Aµ + Aµ. It should also be remembered that this �[ Aµ]corresponds to the gauge-fixing term Fα[A′] = Fα[A′ − A,A] which may be anunusual gauge in the standard approach.

In particular it is convenient to calculate �[ Aµ,Aµ] for Aµ = 0, i.e. restrictingourselves to diagrams with no external Aµ lines. These ‘ Aµ-vacuum’ diagrams arethe 1PI subset of diagrams obtained from W [0,Aµ] given by (8.25), i.e. diagramswith only internal Aµ lines and external Aµ, ghost and other non-gauge field lines.We get

�[Aµ] = �[0,Aµ] (8.32)

A very important fact is that �[0,Aµ] is invariant under the transformation δAµ =Dµ�(x), i.e. it is a gauge-invariant functional of Aµ. This property is maintainedin a perturbative loop-by-loop calculation of �[0,Aµ] (check it). It implies, inparticular, that the divergent terms and the counterterms are manifestly invariantwith respect to the background gauge transformation. Thus the infinities appearingin �[0,Aµ] must take the form of a divergent constant times (Gα

µν)2, where

Gαµν = ∂µAα

ν − ∂νAαµ − gcαβγ Aβ

µAγν

It is then straightforward to show (Problem 8.3) that the renormalization constants

(Aµ)B = Z1/2A Aµ

gB = Zgg

}(8.33)

satisfy

Zg = Z−1/2A (8.34)

and, for instance, the calculation of the β-function in the one-loop approximationsimplifies to the calculation of the diagrams in Fig. 8.2.

It is recommended that the reader derives the Feynman rules necessary for thecalculation of �[0,Aµ] and completes the evaluation of the β-function in the one-loop approximation (Problem 8.3). The background field method is extensivelyused in supergravity theories (Gates, Grisaru, Rocek & Siegel 1983).


Fig. 8.2.

8.3 The structure of the vacuum in non-abelian gauge theories

Homotopy classes and topological vacua

For the moment we shall consider the pure Yang–Mills gauge theory with SU (2)as the gauge group. As we know, the so-called gauge-fixing procedure used inperturbation theory usually leaves some residual gauge freedom. For instance wecan use the gauge A0 = 0 (for the present discussion this is convenient but notnecessary; see, for example, Coleman (1979)) which obviously leaves room forfurther gauge transformations U (r) depending on spatial variables only. Thus thevacuum defined by Gµν = 0 restricts the potentials to ‘pure gauge’ configurations

Ai (r) = −U−1(r)∂iU (r) (8.35)

Further discussion relies on an additional restriction that

limr→∞U (r) = U∞ (8.36)

where U∞ is a global (position-independent) gauge transformation. Equivalently,we assume that all vector potentials fall faster than 1/r at large distances. In pureYang–Mills theory one can argue (Jackiw 1980), but not prove, that physicallyadmissible gauge transformations satisfy (8.36) which ensures that the non-abeliancharge is well defined (see Problem 1.8).

As far as the gauge functions U (r) are concerned the three-dimensional spatialmanifold with points at infinity identified is topologically equivalent to S3 – thesurface of a four-dimensional Euclidean sphere labelled by three angles. Thematrix functions U provide a mapping of S3 into the manifold of the SU (2) gaugegroup. Since any element M in the SU (2) group can be written as

M = a + ib · σσσwhere σσσ s are Pauli matrices and where real a and b satisfy

a2 + b2 = 1

8.3 The structure of the vacuum 283

we conclude that the manifold of the SU (2) group elements is topologicallyequivalent to S3. Thus, gauge transformations U (r) provide mappings S3 → S3.They can be classified into homotopy classes.

Let X and Y be two topological spaces and f0, f1 be two continuous mappingsfrom X into Y . They are said to be homotopic if they are continuously deformableinto each other, i.e. if, and only if, there exists a continuous deformation of mapsF(x, t), 0 ≤ t ≤ 1, such that F(x, 0) = f0(x) and F(x, 1) = f1(x). The functionF(x, t) is called the homotopy. We can divide all mappings of X into Y intohomotopic classes: two mappings are in the same class if they are homotopic. Agroup structure can be defined on the set of homotopy classes.

Let us consider a simple example. Let S1 be a unit circle parametrized by theangle θ , with θ and θ + 2π identified. We are interested in mappings from S1 intothe manifold of a Lie group G. The group of homotopy classes of mappings fromS1 into the manifold of G is called the first homotopy group of G and is denotedby �1(G). Take the mappings from S1 into G ≡ U (1) represented by a set ofunimodular complex numbers z = exp(iu) which is itself topologically equivalentto S1. Thus S1 → S1 and the continuous functions

u(θ) = exp[i(Nθ + a)] (8.37)

form a homotopic class for different values of a and a fixed integer N . One canthink of u(θ) as a mapping of a circle into another circle such that N points of thefirst circle are mapped into one point of the second circle (winding N times aroundthe second circle). Hence the integer N is called the winding number and eachhomotopy class is characterized by its winding number. For this example

�1(U (1)) = �1(S1) = Z

where Z denotes an additive group of integers. We observe that in our example thewinding number N can be written as

N = −i∫ 2π

0

dθ

2π

[1

u(θ)

du

dθ

](8.38)

The mappings of any winding number can be obtained by taking powers of

u(1)(θ) = exp(iθ) (8.39)

Instead of the unit circle S1 we can consider the whole real axis −∞ < x < ∞with the points x = ±∞ identified which is topologically equivalent to S1 (seeProblem 8.1).

This discussion can be generalized by taking X = Sn (the n-dimensionalsphere) or the topologically equivalent n-dimensional cube with all its boundariesidentified and equivalent to some x0 of Sn . The classes of mappings Sn → Sn with


one fixed point f (x0) = y0 form a group called the nth homotopy group of Sm anddesignated by �n(Sm). We have

�n(Sn) = Z (8.40)

i.e. mappings Sn → Sn are classified by the number of times one n-sphere coversthe other. Ordinary space R3 with all points at ∞ identified is equivalent to S3. Aswe have already mentioned the group SU (2) is also equivalent to S3. Thus

�3(SU (2)) = �3(S3) = Z (8.41)

and this together with relations (8.35) and (8.36) implies an infinity of topologicallydistinct vacua for the SU (2) Yang–Mills theory. The same result holds for anarbitrary simple Lie group (for SU (N ) in particular) due to a theorem (Bott1956) which says that any continuous mapping of S3 into G can be continuouslydeformed into a mapping into an SU (2) subgroup of G. Only if the group is U (1),is every mapping of S3 into U (1) continuously deformable into the constant mapf (x) = y0 corresponding to N = 0.

It can be shown that analogously to (8.38) and (8.168) in Problem 8.1 thewinding number for gauge transformations providing the mapping S3 → G isgiven by

N = −(1/24π2)

∫d3x Tr[εi jk Ai A j Ak] (8.42)

where the Ai s are given by (8.35). For prototype mapping with N = 1, seeProblem 8.1.

Physical vacuum

The inequivalence of topologically distinct vacua can be understood as a conse-quence of Gauss’ law

Dabi Gb

0i = 0 (8.43)

To see this let us first recall the canonical formalism. In the Aa0 = 0 gauge the

canonical variables are the Aai s and their conjugate momenta

�ai =

δLδ(∂0 Ai

a)= Ga

0i = ∂0 Aai = −Ea

i (8.44)

The hamiltonian

H =∫

d3x H(x), H = �ai ∂0 Aa

i − L (8.45)


is the energy

H = 1

2

∫d3x (E2

a + B2a ) =

∫d3x �00 (8.46)

Eai = Ga

i0 = −�ai , Ba

i = − 12εi jk Ga

jk (8.47)

When the canonical commutation relations are imposed

[Aai (x, t), Ab

j (y, t)] = [Eai (x, t), Eb

j (y, t)] = 0

[Eai (x, t), Ab

j (y, t)] = iδabδi jδ(x − y)

}(8.48)

the hamiltonian equations give

∂0 Aai = i[H, Aa

i ] = −Eai (8.49)

∂0 Eai = i[H, Ea

i ] = εi jk Dabj Bb

k (8.50)

but Gauss’ law (8.43) is not found. We cannot set Dabi Eb

i to zero as this operatordoes not commute with the canonical variables. Actually, as seen from (8.53), itgenerates infinitesimal space-dependent gauge transformations. Thus, although weobtain a consistent quantum theory, it is different from the desired gauge theory.Gauss’ law can be incorporated into the theory by demanding that the physicalstates are only those which are annihilated by Dab

i Gb0i

Dabi Gb

0i |�〉 = 0 (8.51)

Having (8.51) we can return to our main question and recognize, as the next step,that only gauge transformations which belong to the homotopy class N = 0 canbe built up by iterating the infinitesimal gauge transformations (see Problem 8.2).The generator of these gauge transformations is (see Problem 1.8)

QG =∫

d3x Ga0i Dab

i �b(x) (8.52)

and U = exp(iQG). Integrating by parts and noting that �a(x) vanishes at x →∞for the N = 0 class one gets

QG =∫

d3x [−Dabi Ga

0i�b(x)] (8.53)

Using (8.51) and (8.53) we conclude that such transformations leave the vacuumstate |N 〉 invariant:

T (0)|N 〉 = |N 〉 (8.54)

where T (0) is the unitary transformation corresponding to U = exp(iQG) in theclass N = 0. However, for a gauge transformation in the class N = 1 we have

T (1)|N 〉 = |N + 1〉 (8.55)


Since the states |N 〉 are functionals of Aµ, (8.55) follows from the transformationproperties of the winding number N (A) given by (8.42)

N (U−1n AUn −U−1

n ∂iUn) = N (A)+ n (8.56)

where Un is in the class N = n and Aµ is pure gauge: A = H−1∂µH . Nowthe true vacuum must be at least phase-invariant under all gauge transformationsbecause they commute with observables. Because T is unitary its eigenvalues areexp(−i�), 0 ≤ � ≤ 2π , and the physical vacuum is given by

|�〉 =∞∑

N=−∞exp(iN�)|N 〉 (8.57)

so that

T (M)|�〉 = exp(−iM�)|�〉 (8.58)

The physical vacuum is characterized by � which is a constant of motion andeach |�〉 vacuum is the ground state of an independent sector of the Hilbert space.Indeed, if O is any gauge-invariant operator

[O, UN ] = 0

then

0 = 〈�|[O, UN ]|�′〉 = [exp(−iN�′)− exp(−iN�)]〈�|O|�′〉and

〈�|O|�′〉 = 0 if �′ �= � (8.59)

We want to stress that we have been working in a fixed gauge A0 = 0. Thusthe definitions of the winding number as well as of the topological vacua aregauge-dependent. In our gauge, pure gauge configurations for which E and Bare zero are analogous to infinitely degenerate zero-energy configurations in aquantum mechanical problem with periodic potential. Gauge invariance led us tothe notion of the physical |�〉 vacua as superpositions of the topological vacua (seealso Callan, Dashen & Gross (1976)). As in the quantum mechanical problem, thedegeneracy between different �-vacua is lifted if there exists tunnelling betweentopological vacua. In pure Yang–Mills theories such tunnelling is due to instantons.In the absence of tunnelling, for example, when there are massless fermions in thetheory, the �-vacua remain degenerate and the angle � has no physical meaning.

This interpretation of the |�〉-vacuum is certainly gauge-dependent (Bernard &Weinberg 1977). For instance, one could choose a physical gauge where Aµ isuniquely determined in terms of Gµν with no residual gauge freedom at all. Theclassical vacuum will be unique. However, the physical |�〉-vacuum is, of course,


a gauge-invariant concept. This will become clear in the next subsection where wediscuss the functional integral representation for vacuum-to-vacuum transitions.

�-vacuum and the functional integral formalism

Let us consider a vacuum-to-vacuum transition given by the functional integral

〈0| exp(−iHt)|0〉 ∼∫

DAµ exp

[i∫

d4x (LYM + LG + LFP)

](8.60)

It remains to be understood what one means by ‘vacuum’ for the transition given by(8.60). Firstly, we must pay some attention to the possible boundary conditions onthe fields in the integral (8.60). We are interested in transitions from one classicalvacuum to another so we require that the initial t → −∞ and final t → +∞ fieldconfigurations are of the pure gauge form. Actually, we shall require that gaugepotentials tend to a pure gauge at large distances in all four directions

Aµ −→|xµ|→∞

U−1∂µU (8.61)

For the time being we leave the gauge unspecified.Let us now consider the integral

Q = −(1/16π2)

∫d4x Tr[GµνGµν] (8.62)

where

Gµν = 12ε

µνρσ Gρσ

Q is gauge-invariant and is called the topological charge or the Pontryagin index.It is stationary with respect to any local variation δAµ whether or not the equationof motion is satisfied

δQ ∼ 1

2

∫d4x Tr[Gµν(DµδAν − DνδAµ)]

=∫

d4x Tr[Gµν DµδAν]

= −∫

d4x Tr[DµGµνδAν] +∫

d4x ∂µ Tr[GµνδAν] (8.63)

The first term vanishes from the Bianchi identities (Problem 1.7) and no surfaceterms arise from the second term; Q is a topological invariant. Since (seeProblem 1.6)

−(1/16π2)Tr[GµνGµν] = ∂µKµ (8.64)


where

Kµ = −(1/16π2)εµαβγ Tr[Gαβ Aγ + 23 Aα Aβ Aγ ]

we can write Q as an integral over the surface S at infinity

Q =∫

dSµ Kµ (8.65)

For potentials Aµ satisfying (8.61) Q may be written in terms of U by inserting theasymptotic form of Kµ in (8.65)

Q = −(1/24π2)

∫dSµ εµαβγ Tr[U−1∂αUU−1∂βUU−1∂γ U ] (8.66)

To make contact with the discussion earlier in this section let us take the gaugeA0 = 0. Then from (8.42) and 8.66

Q =∫

d3x K 0∣∣∞−∞ = N (+∞)− N (−∞) (8.67)

Using the remaining gauge freedom we can set N (−∞) = 0. Therefore in thisgauge the topological charge Q can be interpreted as the winding number of thepure gauge configuration to which Ai , tends (we always also assume (8.36) whichis necessary for the integral in (8.67) to be finite) and the functional integral (8.60),with the integral restricted to the Q sector defined by (8.62), as the transitionamplitude between topological vacua |N = 0〉 and |N = Q〉 or more generallybetween |N 〉 and |N + Q〉. In the gauge A0 = 0 we are now ready to construct thefunctional integral corresponding to the |�〉 → |�〉 transition. The result turns outto have the gauge-invariant form for which we are looking. We have

〈�| exp(−iHt)|�′〉 =∑m,n

exp[i(n − m)�] exp[in(�′ −�)]〈m| exp(−iHt)|n〉

= 2πδ(�−�′)∑

Q

exp(−iQ�)〈N + Q| exp(−iHt)|N 〉

∼∑

Q

exp(−iQ�)

∫DAQ exp

[i∫

d4x (LYM + LG + LFP)

]=∫

DAµ exp

(i∫

d4x {LYM + LG + LFP

+ (1/16π2)�Tr[GµνGµν]})

(8.68)

where we have used (8.62). As promised the final result is gauge-independent andit defines the theory having a well-defined vacuum state |�〉 The |�〉-vacuum isdefined by the vacuum expectation values of operators which in turn are determinedby the functional integral with a given value of �. The gauge-dependent notion of


the topological vacua does not appear in this formulation. However, using thepath integral representation (8.68) we can always write the physical vacuum as asuperposition

|�〉 =∫∑

dν exp(iν�)|ν〉

where |ν〉 is an eigenstate of the operator Tr[GµνGµν] with the eigenvalue ν. Thenew term in the lagrangian is a total divergence so it does not influence the classicalequations of motion. Also it does not contribute to the energy–momentum tensor.However, it modifies quantum dynamics which depends on the action. Explicitsolutions for field configurations giving Q = n �= 0 are known (instantons). It isconvenient to work in Euclidean space and to look at Minkowski Green’s functionsas the analytic continuation of the Euclidean ones. The instantons are classicalE = 0 trajectories in Euclidean space-time.†

Fermions must be included in a realistic theory. New interesting aspects thenappear which cannot be considered in detail before discussion of chiral symmetryand anomalies. Let us summarize briefly only the most important facts (see alsoChapter 13). Firstly, take fermions to be massless. Then the �-vacua turn out tobe degenerate: they are the degenerate vacua of spontaneously broken axial UA(1)global symmetry of a theory with massless fermions. The spontaneous breakdownof the UA(1) is the combined effect of the anomaly in the divergence of the axialcurrent and of the existence of field configurations with a non-zero topologicalcharge Q. This mechanism does not imply the existence of the Goldstone bosoncoupled to the gauge-invariant UA(1) current and it offers a prototype solution tothe so-called UA(1) problem.

With massless fermions � changes with chiral rotations and can be rotated away.This is no longer true when fermions are massive since then chiral symmetry is alsobroken explicitly. We face a situation where on one hand the field configurationswith Q �= 0 are welcome to solve the UA(1) problem and on the other hand theexperimentally allowed value of � is surprisingly small as for a strong interactionparameter: from the upper limit on the neutron dipole moment � < 10−19.The �-term violates P and C P conservation so one could argue that we shouldset � = 0 in the strong interaction theory. However, this does not help sinceC P-violating weak interactions renormalize � to a non-zero value. So far noconvincing explanation exists for the smallness of � (Peccei & Quinn 1977,Weinberg 1978, Wilczek 1978, Dine, Fischler & Srednicki 1981). Finally in thepresence of fermions the spontaneous breakdown of the UA(1) and of the chiral

† By a gauge transformation which removes A4 (we are in the gauge A0 = 0, hence in Euclidean space A4 = 0)one can cast the instanton solution into a form which tends to pure gauge configurations satisfying (8.36) andhaving different winding numbers as x4 →+∞ and x4 →−∞. Thus tunnelling takes place and the �-vacuaare not degenerate.


SU (n) × SU (n) are phenomena which are connected to each other by the chiralWard identities based on the spontaneous breaking of the SU (n) × SU (n) and onthe anomalous non-conservation of the UA(1) current.†

The structure of the QCD vacuum briefly discussed in this section is of fun-damental physical importance. However, non-perturbative in nature, it is usuallyneglected in applications of perturbative QCD to short distance phenomena. Onethen simply sets � = 0.

8.4 Perturbative QCD and hard collisions

Historically, the light-cone expansion discussed in Chapter 7 was the originalfoundation for perturbative QCD applications. However, this OPE combinedwith the renormalization group techniques are applicable to a limited number ofproblems and a more general approach based on direct summation of Feynmandiagrams has been widely developed. It can be used for any hard scattering process,not necessarily light-cone dominated, in which there is a large momentum scale q2.If in a given process there are several large momenta O(q2) then we assume theratio of these variables to be fixed as q2 → ∞. This is, for instance, the Bjorkenlimit in the deep inelastic lepton–hadron scattering discussed in Section 7.5.

Parton picture

There are two basic ingredients in applications of perturbative QCD to hadronicprocesses. The first one is to assume the parton picture which provides a connectionbetween QCD and hadrons. No such connection has so far been derived from QCDitself. By the parton picture one means, speaking most generally, the assumptionthat in processes with large momentum transfer q hadrons can be pictured asensembles of partons (quarks and gluons) and that there is no interference betweenthe long time and distance physics responsible for the binding of quarks and gluonsinto hadrons and the short distance effects relevant for the large momentum transferprocess under consideration and occurring at the parton level. The usual argumentto support this conjecture is as follows: view the scattering in a frame in which thehadron (or hadrons) is fast: P → ∞. The time scale of the bound state effects isdilated as P rises. The life-time of a virtual state with quasi-free constituents isP/( E)2 and it becomes long compared to the collision interaction time which isO(1/q0) (here E is the scale of binding effects: E ∼ O(1/r) where r is theradius of the hadron).

The parton picture is implicit in the OPE analysis of the deep inelastic lepton–hadron scattering in Section 7.5 and, for example, in the QCD calculation of the

† This point has been particularly emphasized by Crewther (1979).

8.4 Perturbative QCD and hard collisions 291

total cross section for the e−e+ annihilation into hadrons which is identified withthe total cross section e−e+ → quarks and gluons. In most applications, however,this parton model assumption takes a more concrete form. We assume that thehard scattering cross section can be written as a convolution of soft hadronicwave functions, which are process-independent, with the cross section for the hardsubprocess which involves only partons and can be studied perturbatively. To bespecific let us consider the deep inelastic electron–proton scattering depicted inFig. 7.2. We call P and p the proton and parton momenta, respectively, and usethe so-called light-cone parametrization of the four-momenta

P = (P + m2/4P,O, P − m2/4P)

P = (z P + (p2⊥ + p2

⊥)/4z P, p⊥, z P − (p2⊥ + p2)/4z P)

d4 p = 12(dz/|z|) d2 p⊥ dp2

(8.69)

which defines z as z = (p0 + p3)/2P and is very convenient for this discussion.We assume that in the Bjorken limit

dσ had(P, q) =∑

i

∫dz d2p⊥ dp2 fi (z, p⊥, p2)σ

QCDi (z P, p⊥, p2, q) (8.70)

where σQCDi is the cross section for deep inelastic scattering on the parton ‘i’ as

the target and fi is the four-momentum distribution of partons of type ‘i’ in theproton, i.e. it is related to the square of the soft wave-function. We expect that bothp⊥ and p2 are determined by soft binding effects. In the limit P →∞, p⊥ and p2

finite (infinite momentum frame) we see that p = z P up to corrections O(p⊥/z P).In this limit we envisage the proton as a collection of massless, parallelly movingpartons and fi (z) dz as the average number of partons of type ‘i’ having momentumfraction z, z + dz of the total proton momentum. We shall see shortly that in theinfinite momentum frame (8.70) has a very interesting physical interpretation (see(8.129) and the discussion following it).

Factorization theorem

The second central issue for perturbative QCD calculations is the factorizationtheorem (Ellis et al. 1979, Sachrajda 1983b). It is a generalization of thefactorization encountered in the OPE approach and applies to the dσQCD whichunder the parton model assumption is to be calculated perturbatively as the sumof appropriate quark–gluon Feynman diagrams. The problem is that, as we expectfrom our QED examples, the perturbative expression for dσQCD contains singularcontributions coming from certain regions of phase space in which the quark andgluon momenta are small or parallel. A typical example, but not the only one


(Sachrajda 1983b), is large logarithms like α ln(q2/p2) coming from collinearparton radiation where q2 is the large scale of the process and p2 is the partonmass. The idea of factorization can be illustrated by the identity

[1 + α ln(q2/p2)+ · · ·] = [1 + α ln(M2/p2)+ · · ·][1 + α ln(q2/M2)+ · · ·](8.71)

and the theorem which has been proved to all orders in perturbation theory up topower corrections O(p2/q2) states that, for example, again for the lepton–partondeep inelastic scattering

dσQCDi (p, q) =

∫dy Fi (y, p2/M2) dσi (yp, q2/M2)+ O(p2/M2) (8.72)

where 0 < y < 1 and the Fi are universal process-independent functionscontaining all the logarithms singular in the limit p2 → 0 (mass singularities ormore generally all long distance singularities). The cross sections dσi are finite inthe limit p2 → 0 and M2 is an arbitrary mass introduced to factorize out the longdistance effects of perturbation theory. Using (8.70) and (8.72) the hadronic crosssection can be written as a convolution

dσ had =∑

i

∫fi ∗ Fi dσi =

∑i

∫Fi dσi (8.73)

In the following we explicitly get (8.72) and (8.73) in the first order perturbationtheory. It is, however, absolutely crucial for any relevance of perturbative QCDto hard hadronic processes that the factorization theorem holds asymptotically toall orders in perturbation theory. Then, although we cannot calculate the crosssections dσ had or even dσQCD from first principles, the theory has a clear predictivepower by relating one cross section to another. This may concern the sameprocess at different momentum transfers, for example, scaling violation in the deepinelastic lepton–hadron scattering or two different processes that involve the samedistribution functions Fi . In this second case it is usually convenient to let M2 bethe scale q2 itself and simultaneously be the renormalization scale. For instance,one can compute the cross section for lepton pair production in p p collisions usingdistribution functions defined and measured in the inclusive lepto-production.

In the following we present a sample of lowest order QCD calculations and thenbriefly mention possible strategies for all order extensions. For more details bothtechnical and phenomenological the reader should consult some of the excellent re-view articles (e.g. Buras 1980, Sachrajda 1983a, Reya 1981, Dokshitzer, Dyakonov& Troyan 1980) on this subject.

8.5 Deep e–n scattering 293

8.5 Deep inelastic electron–nucleon scattering in first order QCD (Feynmangauge)

Structure functions and Born approximation

We consider this process in the one-photon exchange approximation. The kinemat-ical variables are defined in Fig. 7.2. The unpolarized cross section for the processeN → eX has the form

dσ = 1

Flux

d3k′

2Ek′(2π)3

∑N

∫ N∏n=1

d3pn

2E pn (2π)3

×(2π)4δ(4)(

k + P − k ′ −∑

n

pn

)e4

q4

1

4

∑pol

|MX |4 (8.74)

where pn are four-momenta of the final particles and X includes all the necessaryspin indices for a given final state (we neglect all particle masses with the exceptionof the initial nucleon mass). Writing

14

∑pol

|MX |4 = LµνWµν (8.75)

where the tensor

Lµν ≡ 12 Tr(/kγµ/k

′γν) = 2(kµk ′ν + kνk ′µ − k · k ′gµν) (8.76)

comes from the lepton vertex we can rewrite the cross section as follows (q ≡k − k ′):

dσ = e4

q4

1

Flux

d3k′

2Ek′(2π)34πLµνWµν (8.77)

where we define

4πWµν ≡∑

N

∫ N∏n=1

d3pn

2E pn (2π)3(2π)4δ(4)

(P + q −

∑n

pn

)Wµν

=∑pol

∫d4x exp(iq · x)〈P|Jµ(x)Jν(0)|P〉 (8.78)

The most general Lorentz and parity-invariant form of the tensor Wµν reads

Wµν = −(gµν−qµqν/q2)W1+[Pµ−qµ(P ·q/q2)][Pν−qν(P ·q/q2)]W2

m2(8.79)

where Wi = Wi (q2, ν), i.e. they are Lorentz scalars. The cross section in thelaboratory frame then reads (Flux = 4(P · k) = 4Em)

dσ

d� dE ′ =m E E ′

π

dσ

d|q2| dν = α2

q4

16E ′ 2

4m[cos2( 1

2θ)W2 + 2 sin2( 12θ)W1] (8.80)


This result follows directly from relations (8.74)–(8.79). From the expres-sion (8.80) we see that the structure functions W1 and W2 defined by (8.75) and(8.78) are real dimensionless functions which are measurable experimentally.

We also introduce another often used notation

F1 = W1, F2 = P · qW2/m2 (8.81)

and express the structure functions F1 and F2 in terms of the tensor Wµν as follows:

gµνWµν = −(n − 1)F1 + 1

2xF2

2xPµPν

P · qWµν = −F1 + 1

2xF2

(8.82)

where x = −q2/2P · q is the Bjorken variable. Relations (8.82) are written forthe general case of n space-time dimensions. Calculating the structure functionsF1 and F2 we also neglect the nucleon mass as we are interested in the kinematicalregion |q2| ∼ s � m2. Structure functions Fi defined by (8.75), (8.79) and (8.81)are free of kinematical singularities when m → 0. We get finally

F1 = 1

n − 2

(−gµν + 2x

PµPν

P · q

)Wµν

1

xF2 = 2F1 + 4x

PµPν

P · qWµν

(8.83)

We want to calculate the structure functions Fi in perturbative QCD under theparton model assumption (8.70).

We assume that partons in a nucleon have some momentum distributionsfi (z) dz, where p = z P . According to (8.70) the lepton–hadron scattering is anincoherent mixture of elementary scatterings

dσ had =∫ 1

0dz∑

i

fi (z) dσQCDi (p = z P, k, k ′) (8.84)

where dσQCDi is the cross section for the lepton scattering on a parton of type ‘i’.

In the actual physical applications of (8.84) we must, of course, remember thathadrons consist of quarks with different flavours and of gluons. Eq. (8.84) is ashort-hand notation for:

in the case of the proton as a target:∑i

fi (z) dσQCDi = [ 4

9(up + up)+ 19(dp + dp)+ 1

9(sp + sp)] dσ q + Gp(z) dσG

(8.85)


in the case of the neutron as a target:∑i


9(un + un)+ 19(dn + dn)+ 1

9(sn + sn)] dσ q + Gn(z) dσG

(8.86)

where the numerical coefficients are the squared electric charges of quarks, qN =qN (z) denotes the momentum distribution of quark q in hadron N and G N (z)denotes the gluon distribution in hadron N . Once the squared electric charges ofquarks are factorized out the dσ q is flavour-independent (and averaged over coloursof the initial and summed over colours of the final partons). The dσG vanishes inthe zeroth order in the strong coupling constant α but contributes in higher orders.

There are several obvious constraints on the qN (z) determined by the quantumnumbers of the nucleon. For instance,∫ 1

0[sp(z)− sp(z)] dz = 0 (strangeness) (8.87)∫ 1

0{ 2

3 [up(z)− up(z)] − 13 [dp(z)− dp(z)]} dz = 1 (charge) (8.88)

etc. Isospin symmetry (p ↔ n, u ↔ d) gives

up(z) = dn(z) ≡ u(z)

dp(z) = un(z) ≡ d(z)

sp(z) = sn(z) ≡ s(z)

Gp(z) = Gn(z) ≡ G(z)

(8.89)

and then for protons∑i


9(u + u)+ 19(d + d)+ 1

9(s + s)] dσ q + G(z) dσG (8.90)

and for neutrons∑i


9(d + d)+ 19(u + u)+ 1

9(s + s)] dσ q + G(z) dσG (8.91)

It is convenient to decompose the quark distribution functions into valence quarkand sea quark distributions

q(z) = qv(z)+ qs(z) (8.92)

where we assume

us = us = ds = ds = ss = ss ≡ S(z) (8.93)


Then for protons∑i


9 uv + 19 dv + 4

3 S(z)] dσ q + G(z) dσG (8.94)

and for neutrons∑i


9 dv + 19 uv + 4

3 S(z)] dσ q + G(z) dσG (8.95)

One also often talks about flavour non-singlet cross sections sensitive only to somecombinations of the valence quark distributions, for example,

dσ ep(x, q2)− dσ en(x, q2) =∫

dz 13 [uv(z)− dv(z)] dσ q(z P, q2) (8.96)

and flavour singlet cross sections sensitive to the distributions 12(qs + qs) = S(z)

and G(z). In the following we calculate dσ q in the zeroth and first orders in thestrong coupling α and study flavour non-singlet hadron structure functions.

We can define the quark structure functions for the elementary subprocess againby (8.79) and (8.81)

W qµν = −

(gµν − qµqν

q2

)Fq

1 (y, q2)

+ 1

p · q

(pµ − qµ

p · q

q2

)(pν − qν

p · q

q2

)Fq

2 (y, q2) (8.97)

where

y = −q2/2p · q = x/z (8.98)

is the Bjorken variable defined for electron–quark scattering. Using (8.84), (8.77)and remembering about the normalizing factor 1/z P0 in dσ q we can express thehadron structure functions in terms of Fq

1,2(y, q2) as follows:

Fhad(x, q2) =∫ 1

0(dz/z)

∑i

fi (z)Fq(x/z, q2) (8.99)

where F(y, q2) stands for F1(y, q2) or (1/y)F2(y, q2).In the Born approximation the quark structure functions are determined by

elastic electron–quark scattering. We have

4πW qµν =

∫dn−1p1

(2π)n−12p01

(2π)nδ(n)(p + q − p1)12 Tr[/pγµ /p1γν] (8.100)

We can also use relation (8.105) to calculate the cut diagram shown in Fig. 8.3.


Fig. 8.3.

Then

W qµν = − 1

4π

∫dn p1

(2π)n(2π)nδ(n)(p1 − p − q)(−2π i)δ+(p2

1)(−i)2 12 Tr[γµi/p1γν /p]

(8.101)(we have used the identity 1 = ∫

[(dn p1)/(2π)n](2π)nδ(n)(p1 − p − q)) whichindeed coincides with (8.100). Using (8.83) we easily get the quark structurefunctions Fi in the Born approximation

(1/y)Fq2 = 2Fq

1 = 2p · qδ(2pq + q2) = δ(1 − y) (8.102)

where y = −q2/2p · q . Indeed, we know already that y = 1 for elastic scattering.We encounter here the so-called Bjorken scaling law: structure functions, which ingeneral can depend on y and q2, depend on y only. This will no longer be the casewhen we include radiative corrections. In terms of the Bjorken variable x definedfor the hadronic process we have

1

x/zFq

2

(x

z

)= 2Fq

1

(x

z

)= δ

(1 − x

z

)(8.103)

where x = −q2/2P · q and p = z P . For the hadronic structure functions we getfinally

(1/x)Fhad2 (x, q2) = 2Fhad

1 (x, q2) =∫

(dz/z)∑

i

fi (z)δ(1 − x/z) =∑

i

fi (x)

(8.104)The Bjorken variable x in (8.104) is defined, as before, in terms of the total nucleonmomentum P , which together with q = k ′ − k, is the kinematic parameter of thereaction. The Bjorken parameter is therefore determined by the specific kinematicconfiguration in our experiment. We arrive at a very interesting conclusion: itfollows from (8.104) that in measuring the hadron structure functions as a functionof x we effectively study the momentum distribution of the hadron constituents.In particular we have learned from experiment this way that half of the nucleonmomentum is carried by gluons. It is clear from calculation of the elastic crosssection (see (8.100)) that our result (8.104) holds up to corrections O(p⊥/z P),


Fig. 8.4.

Fig. 8.5.

where p⊥ is the typical transverse momentum of partons in the hadron. So thecorrections are frame-dependent and the transparent physical interpretation (8.104)of the hadron structure functions and of the Bjorken variable x (for x �= 0) is validin a so-called infinite momentum frame in which P →∞, p⊥ finite.

Deep inelastic quark structure functions in the first order in the strong couplingconstant

We now calculate the dσ q in the first order in αstrong. Figs. 8.4 and 8.5 show thediagrams for the virtual and real corrections. We use the notation of cut diagrams,


Fig. 8.6.

where

Wµν = 1

4π(8.105)

The calculation is performed in the Feynman gauge. The presence for each diagramof the group theoretical factor CF = 3

4 should be kept in mind. It is omitted in mostof the following expressions. All the quantities are for lepton–quark scattering butthe superscript ‘q’ will also be omitted.

Here and in the following we shall be mainly concerned with the quantity

σ = − 1

1 + 12ε

gµνWµν (n = 4 + ε) (8.106)

The term pµ pνWµν in (8.83) is much simpler to calculate.The ladder diagram in Fig. 8.6 gives the following contribution to the cross

section σ

σL = 1

1 + 12ε

αµ−ε

∫d�2

12 Tr[/pγα(/p − l/)γµ/kγ

µ(/p − l/)γ α]

[1

(p − l)2

]2

(8.107)

with

d�2 = dnl

(2π)n2πδ+(l2)

dnk

(2π)n2πδ+(k2)(2π)nδ(n)(q + p − l − k)

(contrary to the calculation of Section 5.5 we do not assume the gluon to be soft).The trace gives

12 Tr[· · ·] = 4(1 + 1

2ε)2W 2[−(p − l)2] = 4(1 + 1

2ε)2 1 − x

xQ2[−(p − l)2]

(8.108)

where W = q + p, Q2 = −q2 and x = −q2/2p · q is the Bjorken variable for the


lepton–quark scattering. Using expression (5.90) for the two-body phase space inthe (p, q) cms we get the following:

σL = α1

8π

1

�(1 + 12ε)

(1 + 12ε)4

(Q2

4πµ2

1 − x

x

)ε/2

×∫

dy [y(1 − y)]ε/2 1 − x

xQ2

[ −1

(p − l)2

](8.109)

with y = 12(1 − z) and z = cos θ , where θ is the polar angle between the three-

vectors l and p. We observe that in the (p, q) cms 2p · l = 2p0l0(1 − z) =p·W (1−z) = (Q2/x)y and the denominator in (8.107) gives a singularity at y = 0,i.e. for l ‖ p and any value of the momentum |l| allowed by the conservation lawq+ p = k+l (4|l|2 = −q2(1−x)/x in cms). This is the so-called mass singularity.We finally get

σL =(

α

2π

)(Q2

4πµ2

)ε/2 (1 + 12ε)

�(1 + 12ε)

�( 12ε)�(1 + 1

2ε)

�(1 + ε)

[(1 − x)

(1 − x

x

)ε/2](8.110)

The pole in ε reflects the mass singularity regularized by working in n = 4 + ε

dimensions. The behaviour near x = 1 is IR behaviour (|l| → 0 when x → 1). Thecut ladder diagram in the Feynman gauge exhibits no IR singularity.† However,we expect that the full cross section for a single photon production is IR divergent.As we know from Section 5.5 in the properly defined physical cross section the IRdivergence is cancelled by virtual corrections to the elastic scattering, i.e.

∫Aδ(1−

x) dx + ∫σ(x, Q2) dx is finite. In order to demonstrate this cancellation explicitly

we write the cross section σ in terms of the so-called regularized distribution. Forany function f (x) we define the distribution f+(x) as follows

f+(x) = f (x)− δ(1 − x)∫ 1

0dy f (y) (8.111)

i.e. we explicitly single out a possibly singular behaviour of the function f (x) asx → 1. We can then write

σL =(

α

2π

)(Q2

4πµ2

)ε/2{(1

ε+ 1 + 1

2γ

)δ(1 − x)

+(

2

ε+ 1 + γ

)[(1 − x)

(1 − x

x

)ε/2]+

}(8.112)

The contribution of the longitudinal term pµ pνWµν can also be easily included.

† Note the difference with Section 5.5: now m = 0.


Fig. 8.7.

The ladder diagram gives(1

xF2

)long

= α

2π3x = α

2π[ 3

2δ(1 − x)+ (3x)+] (8.113)

and the contribution of the rest of the diagrams to pµ pνWµν vanishes.We proceed to calculate cut self-energy corrections. The contribution of the

diagram in Fig. 8.7 reads

σ�c =1

1 + 12ε

αµ−ε

∫d�2

(1

t2

)212 Tr[/pγ µ/tγ α/kγα/tγµ] (8.114)

As before, d�2 is the two-body (l, k) phase space and we work in n = 4 + ε

dimensions. The trace gives 2(1+ 12ε)

22(2p · l)2k · l = 2(1+ 12ε)

22(t2)2 y/(1− x),where y = 1

2(1+ cos θ) and θ is the angle between vectors p and k in the (k, l) cmframe. We see that the propagators are cancelled and we get finally

σ�c = α1

8π

(Q2

4πµ2

1 − x

x

)ε/2

4(1 + 12ε)

1

�(1 + 12ε)

∫ 1

0dy [y(1 − y)]ε/2 y

1 − x

= α

2π

(Q2

4πµ2

)ε/2

(1 + 12ε)

[(1 − x

x

)ε/2 1

1 − x

]�(2 + 1

2ε)

�(3 + 12ε)

(8.115)

In the present case the integration over y is finite: there is no mass singularity. Theresult is, however, divergent for x → 1 (IR divergence). Using the regularizeddistribution we get:

σ�c =α

2π

(Q2

4πµ2

)ε/2{[1

ε+ 1

2(γ − 1)

]δ(1 − x)+ 1

2

(1

1 − x

)+

}(8.116)

Finally we consider the diagrams in Fig. 8.8 and the corresponding cross section

σ�c = 2αµ−ε 1

1 + 12ε

∫d�2

1

t2(p − l)212 Tr[/pγ µ/tγ α/kγµ(/p − l/)γα] (8.117)


Fig. 8.8.

where d�2 is as before. Using γ µ/ab/c/γµ = −2c/b//a − ε/ab/c/ and γ µ/ab/γµ = 4ab +ε/ab/ we evaluate the trace and get

12 Tr[· · ·] = −{4(2p · k)(2p · t − 2t · l)+ 1

2ε4(2p · t)(2k · p − 2k · l)

+ ε Tr[/p/k/t ( /p − l/)] + 12ε

2 Tr[/p/t/k( /p − l/)]} (8.118)

The last two terms do not contribute (one is exactly zero and the second givesε2(−8)(p · l)(k · l), i.e. both propagators are cancelled and there is no divergenceto cancel ε2). In the (k, l) cm frame we have

2p · t = Q2/x, 2p · k = 2p · (t − l) = Q2/x + (p − l)2,

2t · l = 2k · l = (Q2/x)(1 − x)

and

σ�c = 2α1

1 + 12ε

∫d�2 4

[x

1 − x

1

y(1 + 1

2ε)−x

1 − x

(1 + ε

2x

)](8.119)

We see that the cut vertex has (in the Feynman gauge) both IR and mass singularity.Performing the integration over d�2 we obtain the following

σ�c = 2

(α

2π

)(Q2

4πµ2

)ε/2 �(1 + 12ε)

�(1 + ε)

{1

1 + ε

[(2

ε

)2

+ π2

6− 1

]δ(1 − x)

+(

2

ε− 1

)[x

1 − x

(1 − x

x

)ε/2]+

}(8.120)

Final result for the quark structure functions

Virtual corrections to the considered cross section can be easily calculated usingthe results of Chapter 5 for the electron self-energy correction �(p2) and the vertexcorrection �(q2). For instance, the first two diagrams in Fig. 8.4 give together (in


n = 4 + ε dimensions)

σ = −gµνWµν

1 + 12ε

= −1

1 + 12ε

1

4π

∫dnk

(2π)n(2π)nδ(n)(k − p − q)2πδ+(k2)(−i)2

× 12 Tr

[/pγ µ/kγµ

i

/p[−i�(p2 = 0) /p]

]= �(p2 = 0)δ(1 − x) = − α

2π

1

εδ(1 − x) (8.121)

The same result holds for the next two diagrams of Fig. 8.4. Also, with �(p2)

changed into �(q2), each of the vertex correction diagrams can be calculated in asimilar way.

We are now ready to collect all the results (n = 4 + ε everywhere):

Real

σL =(

α

2π

)(Q2

4πµ2

)ε/2{(1

ε+ 1 + 1

2γ

)δ(1 − x)+

(2

ε+ 1 + γ

)×[(1 − x)

(1 − x

x

)ε/2]+

}σ�c =

(α

2π

)(Q2

4πµ2

)ε/2[(1

ε+ 1

2(γ − 1)

)δ(1 − x)+ 1

2

(1

1 − x

)+

]σ�c = 2

(α

2π

)(Q2

4πµ2

)ε/2 �(1 + 12ε)

�(1 + ε)

{[(2

ε

)2

+(π2

6− 1

)]1

1 + εδ(1 − x)

+(

2

ε− 1

)[x

1 − x

(1 − x

x

)ε/2]+

}(

1

xF2

)long

= α

2π[ 3

2δ(1 − x)+ (3x)+]

(8.122)

Virtual

σ� = 2

(− α

2π

)1

εδ(1 − x)

σ� = 2α

2π

{1

ε− 1

ε

�(1 + 12ε)

�(1 + ε)

(Q2

4πµ2

)ε/2

+(

Q2

4πµ2

)ε/2 �(1 + 12ε)

�(1 + ε)

(−1)

1 + ε

×[(

2

ε

)2

+ π2

6

]}δ(1 − x)

(8.123)

The first two terms in σ� correspond to the renormalized �UV (Section 5.3) written


in n = 4+ε dimensions. The final result for (1/x)F2 in the lepton–quark scatteringreads

1

xF2 = CF

(α

2π

)[(2

ε+ γ

)(1 + x2

1 − x

)+ 1 + x2

1 − x

(ln

1 − x

x− 1

)+ 2(1 − x)

+ 1

2

(1

1 − x

)+ 3x

]+

(Q2

4πµ2

)ε/2

+ δ(1 − x)+ O(α2) (8.124)

We now add several important comments on the result obtained to clarify itsphysical sense. As we already know from Section 5.5, the IR singularities cancelbetween real and virtual corrections and they are not present in (8.124). However,this is not the case with mass singularities (due to the decay of a massless particleinto two parallelly moving massless particles) which explicitly reflect themselvesin the final result as poles in ε. We certainly have to cut them off by some physicalparameter before inserting result (8.124) into (8.99) to compare our calculationwith experiment. The trick is to ‘factorize out’ the mass singularities by writingthe result (8.124) in the limit ε → 0 as (see (8.71))

1

xF2(x, Q2) =

∫ 1

0

dz

z�

(x

z, Q2, Q2

0

)1

zF2(z, Q2

0)+ O(α2) (8.125)

where Q20 is an arbitrary parameter and from (8.124)

�(t, Q2, Q20) = δ(1 − t)+ CF

α

2π

(ln

Q2

Q20

)(1 + t2

1 − t

)++ O(α) (8.126)

where the O(α) terms are finite when Q2 →∞. The meaning of (8.125) is clear:the structure function for any Q2 is expressed in terms of the structure functionfor Q2

0 and of the theoretically calculated function �(x/z, Q2, Q20) free of mass

singularity.

Hadron structure functions; probabilistic interpretation

To calculate hadronic structure functions we use the convolution (8.99) and get

1

xFhad

2 (x, Q2) =∫ 1

x

dz

zf (z)

z

xF2

(x

z, Q2

)= 1

x

∫ 1

xdt �(t, Q2, Q2

0)

∫ 1

x/tdz f (z)F2

(x

zt, Q2

0

)(8.127)

where x = Q2/2Pq is the Bjorken variable for the lepton–hadron scattering.Consequently

1

xFhad

2 (x, Q2) =∫ 1

x

dz

z�

(x

z, Q2, Q2

0

)1

zFhad

2 (z, Q20) (8.128)


and interestingly enough we again recover the result (8.125). The effect of theintrinsic quark distribution in the nucleon is included in Fhad

2 (z, Q20). We do not

have to know it theoretically to compare the theory with experiment as far as Q2

evolution of the structure functions is concerned.By introducing an intermediate scale Q2

0 we escape two problems: the intrinsicparton distribution which originates from non-perturbative effects need not beknown theoretically and the perturbative mass singularities are not present inour final formula. They have been absorbed into Fhad

2 (z, Q20), which we take

as the experimentally measured structure function. Our theoretical prediction isthe evolution function �(x/z, Q2, Q2

0) and we do not worry that Fhad2 (z, Q2

0) isactually singular in our calculation since this is due to our ignorance of how toinclude correctly the soft binding effects.

It is also worth observing that the leading Q2 dependence of the result (8.124) forthe quark structure functions and therefore also for the hadron structure functions(8.128) can be interpreted in a very transparent probabilistic parton-like way. Tothis end let us formally rewrite (8.124) as follows

1

xF2(x, Q2) =

∫ 1

0

dz

z

1

zF2(z, Q2)δ

(1 − x

z

)(8.129)

Formally, we obtain an expression for F2 which is the same as (8.104) for thestructure function of a composite object consisting of a set of partons which scatterelastically at large Q2. We are tempted to interpret the quantity (1/z)F(z, Q2)

as the average number of partons, with momentum in the interval zp, (z + dz)p,seen in the initial quark with momentum p by a probe (electron) in the processof deep inelastic scattering with momentum transfer Q2. The theory predicts itsevolution with Q2. Let us assume for the moment that our interpretation is sensible.The hadronic structure function (1/x)Fhad

2 (x, Q2) is then a convolution of theintrinsic parton momentum distribution in a hadron (Q2-independent) and theparton momentum distribution in a parton when probed with large Q2. Technically,the second effect is present in perturbation theory, whereas the first one has itsorigin in non-perturbative binding effects.

Is our interpretation of (8.129) sensible? It is for the leading Q2 dependence.This point will become clearer in the next section when we repeat our calculationin a so-called axial gauge. The reason is that interference diagrams with a gluonemitted or absorbed by the final quark contribute to the final result in the presentcalculation. It turns out, however, that with the proper choice of gauge the leadingeffect comes from the ladder diagram and so our interpretation is possible. Itapplies, however, only to the leading Q2 dependence because to interpret (8.129)in terms of the elastic scattering of a set of partons with momentum distribution(1/z)F2(z, Q2) we have to worry, as before, about corrections O(p⊥/zp). But


now the transverse momentum of a parton is limited only by kinematics and canbe as large as O((Q2)1/2). For the probabilistic interpretation we have not onlyto stay in an infinite momentum frame but also in a limited region of phase space(p⊥ small). This we can do by writing ln(Q2/Q2

0) = ln(εQ2/Q20)+ ln(Q2/εQ2),

where ε is some fixed arbitrarily small number. The first logarithm comes from theregion where p⊥ � (εQ2)1/2 is small and at the same time it gives the leading Q2

dependence correctly (for Q2 →∞). The function

P(z) =(

1 + z2

1 − z

)+

(8.130)

is called the Altarelli–Parisi function. One often considers the Q2 evolution ofmoments of the structure function, e.g.

MN (Q2) =∫ 1

0dx x N−1 F(x, Q2) (8.131)

where again F(x) = F1(x) or F(x) = (1/x)F2(x). From (8.128) and (8.126) onegets

MN =∫ 1

0dx x N−1

∫ 1

0dz∫ 1

0dt �(t, Q2, Q2

0)δ(x − zt)(1/z)F2(z, Q20)

(8.132)or in the leading Q2 dependence approximation

MN (Q2) = MN (Q20)

[CF

α

2πPN ln

(Q2

Q20

)+ 1

](8.133)

where PN is the N th moment of the Altarelli–Parisi function:

PN =∫ 1

0dx x N−1

(1 + x2

1 − x

)+=∫ 1

0dx

1 + x2

1 − x(x N−1 − 1) (8.134)

8.6 Light-cone variables, light-like gauge

In this section we describe briefly the calculation of dσ q given by the diagramsin Figs. 8.4 and 8.5 using the light-cone parametrization of the momenta and thelight-like gauge. To be specific let us consider first the ladder diagram in Fig. 8.6.The gluon propagator in the n2 = 0 gauge is

−i

(gµν − lµnν + lνnµ

l · n

)δαβ (8.135)

8.6 Light-cone variables, light-like gauge 307

and we parametrize the momenta as follows

p = (P + p2/4P,O⊥, P − p2/4P)

l = (z P + l2⊥/4z P, l⊥, z P − l2

⊥/4z P)

t = (y P + (t2 + t2⊥)/4y P, t⊥, y P − (t2 + t2

⊥)/4y P)

q = (ηP + q2/4ηP,O⊥, ηP − q2/4ηP)

y = 1 − z, t2z = −l2⊥

(8.136)

For the gauge four-vector n we take

n = (p · n/2P,O,−p · n/2P) (8.137)

We can interpret (8.136) and (8.137) noticing that P is an arbitrary boost parameterdefining the frame (invariant quantities are P-independent) so that we may start,for example, from the p rest frame (4P2 = p2) and make an infinite boost P →∞in the direction opposite to n. We observe that:

(i) l · n/p · n = z;

(ii) η = −x for p2 = 0 because 2p · q = q2/η;

(iii) 1 − z = x + O(l2⊥(−q2)) because (p − l)2 = −l2

⊥/z and (p − l + q)2 = 0.

The two-body phase space

d�2 = dnl

(2π)n2πδ+(l2)

dnk

(2π)n2πδ+(k2)(2π)nδ(n)(q + p − l − k) (8.138)

can be written as

d�2 = dnt

(2π)n−2δ+((p − t)2)δ+((q + t)2) (8.139)

From

δ+((p − t)2) = δ(−(1 − y)t2/y − t2⊥/y), (p − t)0 > 0

δ+((q + t)2) = δ((y − x)(t2/y − q2/x)− xt2⊥/y), (q + t)0 > 0

}(8.140)

one concludes that

x < y < 1

t2 = y − x

1 − x

q2

x,

q2

x< t2 < 0 (8.141)


Furthermore,

dnt

(2π)n−2= 1

(2π)n−2dt2 dy

2|y| dn−2t⊥ (8.142)

and ∫dn−2t⊥ =

∫tn−3⊥ dt⊥ d�n−2 = π(n−2)/2

�(

12(n − 2)

) ∫ dt2⊥ (t2

⊥)(n−4)/2 (8.143)

So finally, integrating over d(t2⊥/y) by means of the δ((p− t)2), we get (n = 4+ε)

�2 = 1

8π

1

�(1 + 12ε)

∫ 0

q2/xdt2

∫ 1

x

dy

yy

[(1 − y)|t2|

4π

]ε/2

δ

(t2(1 − x)− (y − x)q2

x

)(8.144)

= 1

8π

1

�(1 + 12ε)

1

1 − x

∫ 1

xdy

[(1 − y)(y − x)

x(1 − x)

−q2

4π

]ε/2

(8.145)

For the Feynman part of the ladder diagram, using (8.107) and (8.108) togetherwith (8.144) we recover the previous result (8.110).

The axial part of the ladder diagram

σAL = CF

1

1 + 12ε

αµ−ε

×∫

d�212 Tr

[/pl/

1

/tγµ(/t + /q)γ µ 1

/t/n + /p/n

1

/tγµ(/t + /q)γ µ 1

/tl/

](− 1

l · n

)(8.146)

can be calculated using the expansion

/t = y /p + t2 + t2⊥

2y

/n

p · n+ /t⊥

/q = −x /p + −q2

x

/n

2p · n

(8.147)

and (8.144). After some effort one gets

σAL = CF

(α

2π

)( −q2

4πµ2

)ε/2{2�(1 + 1

2ε)

�(1 + ε)

[(2

ε

)2

− 2

ε+ π2

6

]×δ(1 − x)

(2

ε+ γ

)[2x

1 − x

(1 − x

x

)ε/2]+

}(8.148)


We see that

σ FL + σA

L = CF

(α

2π

)[(2

ε+ γ

)(1 + x2

1 − x

)++ A(ε)δ(1 − x)+ · · ·

]( −q2

4πµ2

)ε/2

(8.149)

where the neglected terms do not have a pole in ε and are polynomials in x andlogarithms in x and 1 − x . Thus in the axial gauge the ladder diagram by itselfgives the leading logarithmic contribution (8.126) to the quark structure function.(Remember that we are interested in the finite x �= 0 region so ln x and ln(1 − x)can be dropped.) We expect therefore that the contribution from the remainingdiagrams with real emissions is non-leading. By explicit evaluation of the axialparts of these amplitudes one gets:

σA�c=(

1

xF2

)long

= 0, σA�c= −σA

L

so that

σ FL + σA

L + σ F�c+ σA

�c+ σ F

�c+ σA

�c= σ F

L + σ F�c+ σ F

�c(8.150)

and (σ F�c+ σA

�c) does not contribute to the leading logarithm, i.e. it is free of mass

singularities. Actually, one can support this conclusion by the following generalargument. The leading logarithm reflecting the presence of mass singularityobviously comes from the integration over t2 in the region εq2 < t2 < 0, i.e.from the region where t and l are almost collinear with p: t ∼= yp. Part of theDirac algebra involves

/tγµ /p ∼= y /pγµ /p = y2pµ /p = 2y

1 − ylµ /p (8.151)

and since in the axial gauge the on-shell gluons have only physical polarization thecontraction with the gluon tensor dµν(l) =∑

λ=1,2 ε∗µ(λ, l)εν(λ, l), l2 = 0 gives

lµ dµν(l) = 0 (8.152)

Actually, one can check that the vertex vanishes linearly in θ , where θ is the gluonemission angle. So the ladder diagram gives∫

d�2 (1/t2)2θ2 ∼∫

dθ θθ2/(θ2)2 ∼ ln θ (8.153)


whereas the cut-vertex diagram gives∫d�2 (1/t2)θ ∼

∫dθ θ(θ/θ2) (8.154)

and hence no logarithm.The poles in ε in the coefficient of the δ(1 − x) are cancelled by the virtual

corrections. Correspondingly the situation with real emissions in the axial gauge,mass singularity is present only in the quark self-energy correction. It is instructiveto calculate the virtual corrections in the light-like gauge and using the light-coneparametrization of the momenta. This we leave as an exercise for the readerlimiting ourselves to the following two remarks. Take, for instance, the self-energycorrection

= −i�

where � can be decomposed as follows

� = A /p + Bp2

2p · n/n (8.155)

with

A = 12 Tr

[/n

2p · n�

](8.156)

A + B = 12 Tr

[/p/n /p

2p · n�

]1

p2(8.157)

The full propagator reads

G(p2) = i

/p −�= i

1 − (A + B)

[/p

p2− B

1 − A

/n

2p · n

]= [1 + (A + B)+ · · ·]

(i/p

p2+ · · ·

)(8.158)

Apart from IR and mass singularities � is also UV divergent, i.e. contains poles inεUV where n = 4 − εUV. The necessary counterterms are of the form

a /p + b//n

The term proportional to /n reflects the pathology of the n2 = 0 gauge because∫d4l

1

(n · l)l2(p − l)2


Fig. 8.9.

should be convergent on dimensional grounds,† and it makes the general proof ofthe renormalizability in this gauge difficult. However, in order-by-order calculationone can check explicitly that such terms cancel out in the full vertex function, seeFig. 8.9.

Our second remark is that an explicit calculation of the virtual corrections usingthe light-cone parametrization of the momenta is called the Sudakov technique. Itconsists of noting that the integral∫

dnl

(2π)n

1

l2 + iε

1

(p − l)2 + iε∼∫

dl2

l2

dz

2|z| dl2⊥(l

2⊥)

(n−4)/2

× z

z(1 − z)(p2 − l2/z)− l2⊥ + iε

(8.159)

can be evaluated by integrating over dl2⊥ first. Going to the complex l2

⊥-plane, sincethere is a pole at

l2⊥ = z(1 − z)(p2 − l2/z)+ iε (8.160)

and the integral over the semi-circle, which is at infinity, vanishes, we can replacethe denominator by δ(z(1 − z)(p2 − l2/z)− l2

⊥).Since in the leading logarithmic approximation, only the ladder diagram con-

tributes to the final result, its probabilistic interpretation is quite natural. In factthis result has the following structure

σL(p, q2) = α

2πCF

∫ q2

p2

dt2

t2

∫ 1

x

dy

y

(1 + y2

1 − y

)+σL(yp, q)

(8.161)where

σL(yp, q) = δ(1 − x/y)

† In the Feynman parametrization such integrals are discussed in Appendix B.


and we have chosen to regularize the mass singularity with p2 �= 0. This coincideswith (8.129) and

Pq→q(y)

(1 + y2

1 − y

)+

can be interpreted as the variation of the probability density per unit t2 of findinga quark in a quark with fraction y of its momentum. Of course, our previousdiscussion concerning the hadron structure functions, (8.127)–(8.129), is still valid.

8.7 Beyond the one-loop approximation

Since αstrong ∼ O(1) the relevance of perturbative QCD to the description of theactual experimental data often relies on our ability to go beyond the one-loopapproximation. In addition we must also include in considerations the singlet andthe gluon structure and fragmentation functions. This vast area of technical andphenomenological activity goes far beyond the scope of the present book and canby itself be a subject of an extended monograph. Let us only briefly indicate thetechniques that have been developed to include higher order effects in αstrong. Oneapproach is based on the OPE and on the RGE for the Wilson coefficient functions(Chapter 7). Another one, applicable to a wider class of problems, consists ofsumming infinite sets of Feynman diagrams by cleverly picking out the leadingcontribution, the next-to-leading one and so on (Gribov & Lipatov 1972, LlewellynSmith 1978, Dokshitzer, Dyakonov & Troyan 1980). For the regions x �= 0and x �= 1, i.e. neglecting ln x and ln(1 − x) terms as compared to ln(q2/p2)

terms, the summation of the so-called leading logarithms αn lnn(q2/p2) and alsoof the next-to-leading αn lnn−1(q2/p2) terms has been explicitly performed andcompared, whenever possible, with the OPE. It is very interesting that theseresults can also be given a probabilistic interpretation. In the leading logarithmapproximation it is concisely formulated by means of the Altarelli–Parisi equations(Altarelli & Parisi 1977).

Further very interesting and more recent progress consists of extending the QCDperturbative techniques to processes where there are two types of large logarithm(for example, Dokshitzer, Dyakonov & Troyan (1980), Parisi & Petronzio (1979))and to the so-called doubly-logarithmic region of the structure and fragmentationfunctions (x ∼= 0). In the context of QCD pioneered by Furmanski, Petronzio &Pokorski (1979) and Bassetto, Ciafaloni & Marchesini (1980) and fully clarified byBassetto, Ciafaloni, Marchesini & Mueller (1982), this extension is, for instance,important for studying new particle production in hard collisions in very highenergy accelerators.

8.7 Beyond the one-loop approximation 313

Fig. 8.10.

Let us be a little bit more explicit concerning the summation of the leadinglogarithm terms αn lnn(q2/p2) generated by mass singularities in the quark struc-ture function. It turns out that the dominant diagrams in the axial gauges arethe generalized ladder diagrams, Fig. 8.10, i.e. ladder diagrams with vertex andself-energy insertions. The dominant region of integration is where

|p2| � |t21 | � |t2

2 | � · · · � |t2n | � |q2|

Furthermore, it can be shown that the generalized ladder can be replaced bya ladder with the coupling constant α(t2

i ) at each vertex. Thus in the leadinglogarithm approximation the cross section is a generalization of (8.161) and reads

σ nL =

∫ |q2|

p2

dt2n

t2n

CFα(t2

n )

2π

∫ |t2n |

p2

dt2n−1

t2n−1

CFα(t2

n−1)

2π· · ·

×∫ |t2

2 |

p2

dt21

t21

CFα(t2

1 )

2π

∫ 1

xdy1 Pq→q(y1)

∫ y1

x

dy2

y1Pq→q

(y2

y1

)· · ·

×∫ yn−2

x

dyn−1

yn−2Pq→q

(yn−1

yn−2

)∫ yn−1

x

dyn

yn−1Pq→q

(yn

yn−1

)δ(yn − x)

1

yn

(8.162)

Using (8.162) we can easily calculate the moments of (1/x)F2. First we note that(CF

2π

)n ∫ |q2|

p2

dt2n

t2n

α(t2n )

∫ |t2n |

p2

dt2n−1

t2n−1

α(t2n−1) · · · →

1

n!(CF/2)n

×∫ |q2|

p2

dt2n

t2n

α(t2n )

∫ |q2|

p2

dt2n−1

t2n−1

α(t2n−1) · · · =

1

n!

[ln

ln(−q2/�2)

ln(−p2/�2)

]n(−CF

b1

)n

(8.163)


where α(t2) = −2π/[b1 ln(t2/�2)],−b1 = 112 − 1

3 n f . Changing variables

z1 = y1, z2 = y2/y1, . . . , zn = yn/yn−1

z1z2 . . . zn = yn

we get∫ 1

0dx x N

∏i

∫dzi Pq→q(zi )

1

z1 . . . znδ(x − z1 . . . zn) =

∏i

∫ 1

0dzi zN−1

i Pq→q(zi )

(8.164)Therefore, finally

MN =∫ 1

0dx x N 1

xF2(x, q2)

=∑

n

1

n!

{[ln

ln(−q2/�2)

ln(−p2/�2)

][−CF

b1

∫ 1

0dz zN−1 Pq→q(z)

]}n

=[

ln(−q2/�2)

ln(−p2/�2)

]−(CF/b1)∫ 1

0 dz zN−1 Pq→q (z)

(8.165)

and

MN (Q2)

MN (Q20)=[

ln(Q2/�2)

ln(Q20/�

2)

]γ 0N /b1

, Q2 = −q2 (8.166)

where

−γ 0N = CF

∫ 1

0dz zN−1

(1 + z2

1 − z

)+= −2

3

[1− 2

N (N + 1)+4

N−2∑j=2

1

j

](8.167)

The same result holds, of course, for the moments of the hadron structure functionand it coincides with (7.124) obtained using the OPE. We see that γ 0

N are the g2-coefficients in the expansion of the anomalous dimensions of the operators in (7.99)in the coupling constant.

Comments on the IR problem in QCD

The Bloch–Nordsieck theorem in QED states that IR divergences cancel forphysical processes, i.e. for processes with an arbitrary number of undetectable softphotons. In QCD it has been demonstrated that IR divergences cancel for processeswith none or one coloured particle in the initial state (for a review see, for example,Sachrajda (1983b)). However, it has been shown by means of an explicit examplethat such a cancellation does not occur for processes with two coloured particles inthe initial state even if an average over different colour states is taken. For instance,in the process qq → l+l−X these divergences arise at the two-loop level. Of course,

Problems 315

in physical processes coloured partons never appear as free on-shell particles andthese divergences are regulated by a mass-scale characteristic of the hadronic wavefunction which is of the order of the inverse hadronic radius. However, residuallarge logarithms remain and could, in principle, spoil the factorization propertydiscussed in Section 8.4 which states that all long distance effects can be absorbedinto the quark and gluon distribution functions and is crucial for the sensible useof perturbative QCD. Fortunately enough it has been shown that such large IRlogarithms are suppressed by at least one power of Q2 and asymptotically thefactorization property is still maintained.

In the context of the IR problem in QCD it is worth recalling the Kinoshita–Lee–Nauenberg (KLN) theorem (Kinoshita 1962, Lee & Nauenberg 1964). TheKLN theorem is a mathematical theorem saying that, as a consequence of unitarity,transition probabilities are finite when the sum over all degenerate states (finaland initial) is taken. This is true order-by-order in perturbation theory in barequantities or if the minimal subtraction renormalization is used (to avoid IR ormass singularities in the renormalization constants). The physical meaning of thistheorem seems, however, to be unclear. In QED the cancellation of IR singularitiesoccurs separately in the final state and there is no need to invoke the KLN theorem.In non-abelian gauge theories this does not happen but the initial state is determinedby the non-perturbative confinement effects and the relevance of the KLN theoremis doubtful.

Problems

8.1 (a) Consider mappings of the real axis with the end-points identified into a set ofunimodular complex numbers z = exp[iα(x)]. Check that the winding numbercan be written as

N = 1

2π

∫ ∞

−∞dx

[ −i

α(x)

dα(x)

dx

](8.168)

Check that N = 1 for α(x) = exp[iπx/(x2 + a2)1/2].(b) For S3 → SU (2) the transformation

Aµ → exp(− 12 i�aσ a)(Aµ + ∂µ) exp( 1

2 i�aσ a)

with �a(x) ≡ 0 or �a(x) continuously deformable to �a(x) ≡ 0, belongs toN = 0. Clearly, then �a(∞) = 0. Check that with

�a(x) = 2πxa/(x2 + a2)1/2

or, with �a continuously deformable to it, U (x) belongs to N = 1 class. Whatis U∞? Can such a transformation be built up from infinitesimal transformations�a(x) → δ�a(x) for all x?


8.2 Show that gauge transformations which belong to the homotopy class N �= 0 cannotbe built up by iterating the infinitesimal gauge transformations. For this assume that a‘large’ (N �= 0) gauge transformation

U = exp(− 12 i�aσ a)

where �a(x) approaches a non-zero limit at spatial infinity is given by the operator

exp(iQG)A exp(−iQG) = U−1AU −U−1∇∇∇U

where

exp(iQG) = exp

(−i∫

d3x Diab�b ·�i

a

)Applying it to

W (A) =∫

d3x K 0

where K 0 is given by (8.64), show that

exp(iQG)W (A) exp(−iQG)W (A)

contradicting the correct result

exp(iQG)W (A) exp(−iQG) = W (U−1AU −U−1∇∇∇U ) = W (A)+ N

(Use δW (A)/δAia = (1/8π2)Ba

i .)

8.3 Derive the Feynman rules for calculating the 1PI Green’s functions generated by�[0,Aµ] defined in Section 8.2. Use the shifted action S[Aµ + Aµ], the gauge-fixingterm (8.24) and the Faddeev–Popov term obtained from (3.15) and (3.39) with Fα →Fα[Aµ,Aµ] and δAµ given by (8.22). Prove (8.34) and complete the calculation ofthe β-function in Yang–Mills theories in the one-loop approximation (Abbott 1982).

9

Chiral symmetry; spontaneous symmetry breaking

Chiral symmetry plays an important role in the dynamics of fermions coupled togauge fields. Historically, its importance for particle physics originates in the ideathat SU (2)L × SU (2)R is an approximate symmetry of the strong interactions,realized as spontaneously broken symmetry with pions being Goldstone bosons inthe symmetry limit. Much of the progress of particle physics since 1960 is relatedto understanding the phenomenological aspects of spontaneous and explicit chiralsymmetry breaking. The successes of gauge theories in describing the fundamentalinteractions further stimulate interest in chiral symmetry and this is for at leasttwo reasons: one may hope firstly to understand the underlying mechanism ofspontaneous chiral symmetry breaking in the strong interactions and in gaugetheories in general and, secondly, to find the dynamical theory of the fermion massmatrix which might eventually lead to the true theory of fundamental interactions.This chapter serves as an introduction to the subject of chiral symmetry and totechniques used in its exploration.

9.1 Chiral symmetry of the QCD lagrangian

We begin our discussion by considering the QCD lagrangian with two quarks u andd in the theoretical limit in which the quark mass parameters in the lagrangian areput equal to zero. In fact, the masses of the light quarks are small in a sense whichwill be specified later on and we shall assume in the following that the theory withthe quark masses neglected is a correct first approximation.

Consider the general massless Dirac lagrangian of fermion fields �ri coupled togauge fields Aα

µ

L =∑

r

n(r)∑i

�ri iγµDµ�ri , � = �(x) (9.1)

317

318 9 Chiral symmetry; spontaneous symmetry breaking

where

Dµ = ∂µ + ig Aαµtαr

and the index α runs over the generators of the gauge group, the matrices tαrrepresent these generators in the representation r of the gauge group to which thefermions are assigned and the index i runs over all n(r) flavours belonging to thesame representation r . For QCD with two flavours r ≡ 3, i = 1, 2 and �i (we nowomit the subscript r ) denotes a vector in the colour space. In this case lagrangian(9.1) is invariant under the general unitary transformation in the two- dimensional

flavour space. Introducing � =(�1

�2

)we can write

L = �

(iD/

iD/

)� (9.2)

which is invariant under SU (2)×U (1) transformations

� → exp(−iαa 12σ

a)�

� → � exp(+iαa 12σ

a)

}(9.3)

where a = 0, 1, 2, 3 and σ a are Pauli matrices for a = 1, 2, 3 and σ 0 = 1. Thecorresponding conserved currents are

Vaµ = �γµ

12σ

a� (9.4)

and the SU (2)×U (1) invariance gives the isospin and the baryon number

Qa =∫

d3x �(x)γ012σ

a�(x), a = 0, 1, 2, 3 (9.5)

conservation.However, the massless theory (9.2) also has another global symmetry, involving

γ5

� → exp(−iαa 12σ

aγ5)�

� → � exp(−iαa 12σ

aγ5)

}(9.6)

where a = 0, 1, 2, 3. Indeed, since {γ5, γµ} = 0 the exponentials again cancelwhen (9.6) is introduced into lagrangian (9.2). The axial currents read

Aaµ = �γµγ5

12σ

a� (9.7)

9.1 Chiral symmetry of the QCD lagrangian 319

and the axial charges†

Qa5 =

∫d3x �(x)γ0γ5

12σ

a�(x) (9.8)

together with the charges (9.5) form the chiral SUL(2)× SUR(2)×U (1)×UA(1)algebra (as can be seen using (9.5) and (9.8) and the equal-time canonicalcommutation relations between the fields)

QaL = 1

2(Qa − Qa5), Qa

R = 12(Qa + Qa

5), a = 0, 1, 2, 3

[QaL,R, Qb

L,R] = iεabc QcL,R, a, b, c = 1, 2, 3

[QaL,R, Qb

R] = 0, a, b = 0, 1, 2, 3

(9.9)

(for completeness: [Qa, Qb] = [Qa5, Qb

5] = iεabc Qc; [Qa5, Qb] = iεabc Qc

5). Animportant property of the generators Qa

L and QaR is that they transform into each

other under the parity operation: P QaR,L P−1 = Qa

L,R; hence the name chiraltransformations.

Before we proceed to explore the chiral symmetry of the massless QCDlagrangian (9.2) we introduce some useful notation. Any spinor field � can bedecomposed into ‘chiral’ fields‡ as follows

� = �L +�R = 12(1 − γ5)� + 1

2(1 + γ5)� = PL� + PR� (9.10)

where

P2L + P2

R = 1, PL PR = 0

and the lagrangian (9.2) can then be rewritten as

L = �LiD/ 1l�L + �RiD/ 1l�R (9.11)

(1l is unit matrix in the flavour space).Chiral symmetry of the lagrangian (9.11) means invariance under the transfor-

mations

� iL → � ′ i

L = (UL)ij�

jL

� iR → � ′ i

R = (UR)ij�

jR

}(9.12)

where indices i, j are flavour indices and the unitary transformations UL and UR

readUL = exp(−iαa

LT aL )

UR = exp(−iαaRT a

R )

}(9.13)

† Charges may be formal in the sense that the integral over all space may not exist. Even so, their commutatorsexist provided we first commute under the space integrals and then do the integrations. This remark is relevantfor the case of spontaneous symmetry breakdown.

‡ Decomposition (9.10) has a clear interpretation in terms of the free particle solutions of the massless Diracequation. This is discussed in Appendix A.


with T aL = T a ⊗ PL and T a

R = T a ⊗ PR. The matrices T form a representation ofthe corresponding generators Q in the space of fermion multiplets �. We see thatmultiplet �L(�R) is a singlet with respect to SUR(2)(SUL(2)) transformations. Atthis level we should also conclude that because of the U (1) × UA(1) invariancethe fermion numbers of �R and �L (corresponding to U (1) ± UA(1) charges)are separately conserved. The problem of the UA(1) symmetry will be brieflydiscussed in Chapter 13. For the time being we will be concerned with the chiralSU (2) × SU (2) only. Some useful properties of the chiral fields �L and �R aregiven in Appendix A.

In many problems we work with one type of chiral field �R or �L only. Forinstance in ‘chiral’ gauge theories such that �R and �L transform differently underthe gauge group (for example, in grand unification schemes) only �L (or only �R)fields can be placed in irreducible multiples of the gauge group. Indeed, �Rγµ�L

currents vanish and we would not be able to introduce symmetry operationsbetween �R and �L. The description in terms of, for example, the left-handedfields can be obtained if we replace �R by their charge conjugate partners

(�R)C = (�C)L =

df�L (9.14)

Then we can formulate our theory in terms of �L and �L. Another notation worthmentioning is the one using the two-component Weyl spinors. We refer the readerto Appendix A for further details on both subjects.

9.2 Hypothesis of spontaneous chiral symmetry breaking in stronginteractions

We have assumed that the massless QCD with its chiral SUL(2) × SUR(2) globalsymmetry is a good approximation to the real world. Now let us imagine in additionthat the physical vacuum defined by the minimum of the expectation values of thehamiltonian 〈0|H |0〉 = 〈H〉min is invariant under the chiral transformations, i.e.

QaR|0〉 = Qa

L|0〉 = 0 (9.15)

We can invoke Coleman’s theorem (see, for example, Itzykson & Zuber (1980),p. 513): ‘a symmetry of the vacuum is a symmetry of the world’ or

Qi |0〉 = 0 ⇒ [Qi , H ] = 0 (9.16)

to conclude that the physical states in the spectrum of H can be classified accordingto the irreducible representations of the chiral group generated by Qa

R,L (as canbe seen by constructing them from the vacuum by means of the field operatorsand then using the properties (9.16) and the transformation properties of the field

9.2 Spontaneous chiral symmetry breaking 321

operators under the chiral group). It is easy then to see that all isospin multipletswould have to have at least one degenerate partner of opposite parity. Let the state|�〉 be an energy and parity eigenstate:

H |�〉 = E |�〉 and P|�〉 = |�〉Since QL and QR commute with the hamiltonian we also have

H QL|�〉 = E QL|�〉 and H QR|�〉 = E QR|�〉In addition

P QLR|�〉 = P QL

R P+P|�〉 = QRL|�〉

and therefore the state

|� ′〉 = 1√2(QR − QL)|�〉

degenerate with |�〉 is also a parity eigenstate

P|� ′〉 = −|� ′〉The presence of parity degenerate states is not a general feature of the knownhadron spectrum. Therefore, if we wish to maintain our assumption about masslessQCD being a good approximation to the strong interaction we must give upassumption (9.15). Instead we assume that the physical vacuum is not invariant(but the hamiltonian remains invariant) under the full chiral group and

Qa5|0〉 �= 0, Qa|0〉 = 0 (9.17)

One then talks about spontaneous breakdown, or Nambu–Goldstone realization(Nambu 1960, Goldstone 1961), of the chiral symmetry SUL(2) × SUR(2) intoisospin symmetry SU (2) generated by Qa . Of course, given a theory like masslessQCD the choice between (9.15) and (9.17) is no longer free for us. The theory itselfshould, in principle, tell us whether the symmetry or part of it is spontaneouslybroken or not. Unfortunately, the problem of calculating the spontaneous chiralsymmetry breaking in QCD has not yet been satisfactorily solved. It is theninteresting to begin as we do: assume the spontaneous breakdown (9.17) andexplore its consequences in the hope of finding support for the assumption.

The most important consequence of the spontaneous symmetry breakdown issummarized in Goldstone’s theorem: if a theory has a global continuous symmetryof the lagrangian which is not a symmetry of the vacuum there must be a massless(Goldstone) boson, scalar or pseudoscalar, corresponding to each generator (withits quantum numbers) which does not leave the vacuum invariant. The proof ofthe theorem will be given later. A very simple heuristic argument is as follows:


if Qa5|0〉 �= 0 then there is a state Qa

5|0〉 degenerate with the vacuum† because[Qa

5, H ] = 0. By successive application of Qa5 one can construct an infinite number

of degenerate states. A massless boson with internal quantum numbers, spin andparity of Qa

5 must exist to account for the degeneracy.Very plausible candidates for Goldstone bosons in the strong interactions with

spontaneously broken chiral symmetry are pseudoscalar mesons. Restrictingourselves to the SU (2) × SU (2) case we see that the triplet of pions has the rightquantum numbers for such an interpretation. That they are not exactly massless(but nevertheless much lighter than other hadrons) is attributed to an explicit chiralsymmetry breaking by the quark mass terms in the lagrangian.

Another consequence of spontaneous chiral symmetry breaking which agreeswith the experimental data is the Goldberger–Treiman relation. This is based onthe identification of the axial current of the two-flavour QCD with the axial currentof the weak interactions for the first generation of quarks. This identification isexplicit in the SU (2) × U (1) Glashow–Salam–Weinberg theory of weak interac-tions. The derivation of the Goldberger–Treiman relation in the limit of exact chiralsymmetry (with quark masses and pion mass neglected) which is spontaneouslybroken, then goes as follows: consider the matrix element

〈0|Aaµ(x)|πb(q)〉 = exp(−iqx)〈0|Aa

µ(0)|πb(q)〉 (9.18)

where, from Lorentz covariance,

〈0|Aaµ(0)|πb(q)〉 = i fπqµδ

ab (9.19)

The constant fπ is called the pion decay constant and π(1) = (1/√

2)(π+ +π−), π(2) = −(i/

√2)(π+ − π−), π3 = π0 (therefore, the π± decay constant

is√

2 fπ). In the Born approximation for the electroweak decays (see Chapter 12)the pion decay, π → lν, amplitude is proportional to the matrix element (9.18)(we neglect the Cabbibo angle). We assume the axial charge does not annihilatethe vacuum, therefore fπ �= 0. In our approximation the axial current is conserved:∂µAa

µ(x) = 0 and from (9.18) and (9.19) we get

〈0|∂µAaµ(x)|πb(q)〉 = exp(−iqx) fπm2

πδab (9.20)

This is Goldstone’s theorem: fπ �= 0 therefore m2π = 0. Next consider nucleon

matrix elements of the axial current (β-decay). Assuming C invariance and isospin

† This interpretation requires some caution (Bernstein 1974) since Qa5 |0〉 is not a normalizable state. It means

that U = exp(−iαa Qa5) is not a unitary operator. Nevertheless, expressions like U AU+ can be meaningful.

If the exponentials are expanded, the resulting commutators may be well defined if we first commute currentsand then integrate over space.

9.2 Spontaneous chiral symmetry breaking 323

invariance one has

〈N(p′)|A−µ(x)|P(p)〉 = exp(iqx)u(p′) 1

2σ−[γµγ5gA(q

2)+ qµγ5h(q2)]u(p)(9.21)

where

σ− = (1/√

2)(σ 1 − iσ 2), q = p′ − p

Current conservation implies

2mNgA(q2)+ q2h(q2) = 0 (9.22)

where mN is the nucleon mass. For q2 = 0 this relation can be satisfied either forgA(0) = 0 or if h(q2) has a pole for q2 = 0. If pions are Goldstone bosons theycontribute such a pole as can be seen from (9.23)

= √2gπN(i/q2)i

√2 fπqπ uγ5u exp(iqx) (9.23)

where gπN is the pion–nucleon coupling constant with the same normalizationconvention as for fπ . Hence

mNgA(0) = gπN fπ (9.24)

which is the Goldberger–Treiman relation derived from the assumption of sponta-neously broken chiral symmetry of strong interactions. One can now ask whetherthe relation (9.24) is a good approximation to the real world where m2

π �= 0.Experimentally, from πN scattering g2

πN/4π = 14.6, from (physical) π → lν| fπ | = 93 MeV, from β-decay gA(0) = 1.24 and relation (9.24) happens to besatisfied within 7%. This agreement supports our main assumption: massless QCDwith spontaneously broken chiral symmetry is a good approximation for the stronginteractions.

Relating physical quantities to the results obtained in the exact symmetrylimit m2

π = 0 is called the PCAC (partially conserved axial vector current)approximation. Originally, the PCAC approximation has been formulated inanother, equivalent, way as an operator equation

∂µAaµ(x) = m2

π fππa(x) (9.25)

where m2π is the physical pion mass, πa(x) is the pion field normalized as follows:

〈πa(p)|πb(x)|0〉 = δab exp(ipx). Relation (9.25) is an identity on the pion mass-shell, as can be seen from (9.20), and it is assumed that the pion field so defined is


a good interpolating field off-shell at p2 = 0. For a more extensive discussion ofthe derivation of the Goldberger–Treiman relation based on relation (9.25) and itsconnection to the chiral derivation see, for example, Pagels (1975).

9.3 Phenomenological chirally symmetric model of the strong interactions(σ -model)

As we have already mentioned, calculating spontaneous chiral symmetry breakingin QCD is a difficult problem which has not yet been solved. It has, therefore,proved instructive in several aspects to study a phenomenological chirally sym-metric field theory model of the strong interactions, the so called σ -model. In thismodel the scalar fields are introduced as elementary fields and their interactionsare arranged to produce the spontaneous breakdown of the symmetry.

The elementary fields in the σ -model are: the nucleon doublet, massless to startwith so in fact we have two doublets NR and NL transforming like (1, 2) and (2, 1)under SUL(2)× SUR(2); the triplet pion field πππ(0−) and σ ′-a scalar 0+ meson.

The postulated lagrangian is as follows

L = N i/∂N + gN (σ ′ + iσσσ ·πππγ5)N + 12 [(∂µπππ)2 + (∂µσ

′)2]

− 12µ

2(σ ′2 +πππ2)− 14γ (σ ′2 +πππ2)2 (9.26)

where

πππ = (π1, π2, π3), σσσ = (σ 1, σ 2, σ 3)

This is the usual pseudoscalar pion–nucleon coupling. The fourth field σ ′ has beenintroduced to get a chirally symmetric theory. We rewrite

N i/∂N = NRi/∂NR + NLi/∂NL

N (σ ′ + iπππ · σσσγ5)N = NL(σ′ + iπππ · σσσ)NR + NR(σ

′ − iπππ · σσσ)NL

}(9.27)

to see that for chiral invariance of the lagrangian the (σ + iπππ · σσσ) must transformas the (2, 2) representation of the SU (2) × SU (2) (remember that 2∗ ≡ 2). Thiscorresponds to the following transformations (see Appendix E):

under vector SU (2), i.e. isotopic spin group

πππ → πππ + δααα ×πππ

σ ′ → σ ′

under axial vector rotationsπππ → πππ + δααασ ′

σ ′ → σ ′ − δααα ·πππ

(9.28)

9.3 σ -model 325

(δααα is an infinitesimal vector in the isospin space). Chiral symmetry impliesconserved axial and vector currents. They read

Vaµ(x) = N (x)γµ

12σ

a N (x)+ [πππ(x)× ∂µπ(x)]a

Aaµ(x) = N (x)γµγ5

12σ

a N (x)+ σ ′(x)∂µπa(x)− πa(x)∂µσ ′(x)

}(9.29)

Constructing then charges Qa, Qa5, QL, QR and using canonical commutation

relations for fields one gets

[QaLR, σ ′] = ∓ 1

2 iπa

[QaLR, πb] = 1

2 iεabcπ c ± 12 iδabσ ′

(9.30)

Note (Appendix E) that in general if a field multiplet � transforms as �i → �′i =

(δi j − iδαaT ai j )� j under linear transformations generated by generators Qa then,

since on the other hand the operator relation

�i → �′i = exp(iαa Qa)�i (−iαa Qa)

∼= �i + iδαa[Qa,�i ] (9.31)

holds, it must be that

[Qa,�i (x)] = −T ai j� j (x) (9.32)

(matrices T a form a representation of generators Qa).It is now necessary to establish the correct ground state of the model. The

vacuum is defined by the minimum of the expectation values of the hamiltonian.As we know, for example, from the effective potential formalism, in the treeapproximation the vacuum expectation values of the fields are equal to the valuesof the classical fields in the minimum of the potential V (σ ′,πππ). We can rewrite thepotential as

V (σ ′,πππ) = − 14λ(σ

′2 +πππ2 + µ2/λ)2 (9.33)

to see that for µ2/λ > 0 the minimum occurs for σ ′ = πππ = 0, whereas forµ2/λ < 0 it exists for

σ ′ 2 +πππ2 = |µ2/λ| (9.34)

With this second choice the model accounts for the spontaneous breakdown ofchiral symmetry. The physical vacuum is chosen (defined) from the degenerateground states given by the condition (9.34). In particular we would like to preserveunbroken isospin symmetry. The physical vacuum must then satisfy (9.34) andalso

Qa|0〉 = 0, Qa5|0〉 �= 0 (9.35)


These conditions are satisfied with the choice

〈0|πππ |0〉 = 0, 〈0|σ ′|0〉 = −(|µ2/λ|)1/2 ≡ v (9.36)

Indeed the vacuum must be an isospin singlet (hence 〈0|πππ |0〉 = 0) but not a chiralsinglet (hence 〈0|σ ′|0〉 �= 0, where σ ′ belongs to the chiral doublet). Formally,one can see that 〈0|σ ′|0〉 �= 0 implies Qa

5|0〉 �= 0 (and vice versa) by taking thevacuum expectation value of the commutator [Qa

5, πb] = −iδabσ ′. Another way

is to consider 〈0| exp(iαa Qa5)σ

′ exp(−iαa Qa5)|0〉 to conclude that generators of the

transformations which do not leave 〈0|σ ′|0〉 invariant are spontaneously broken.It is worth stressing at this point that in our discussion of the σ -model we havefixed at the very beginning the physical interpretation of the group generators andof the fields. In the basis so defined the choice of the physical vacuum among thedegenerate ground states given by the condition (9.34) is no longer arbitrary if wewant to keep the isospin unbroken. However, we could also proceed in anotherway: define any of the a priori equivalent degenerate ground states as the physicalvacuum, find the broken and unbroken generators of the theory, give to them andto the fields the desired physical interpretation. The pattern of the spontaneoussymmetry breaking from such a general viewpoint will be discussed in Section 9.5.

In view of (9.36) we must, to ensure the orthogonality of the vacuum to theone-particle state, define the physical σ field as

σ = σ ′ − v (9.37)

The lagrangian (9.26) can now be rewritten in terms of the field σ

L = N i/∂N + gN (σ + iσσσ ·πππγ5)N + gv N N + 12 [(∂µπππ)2 + (∂µσ )2]

− |µ2|σ 2 − 14λ(σ

2 +πππ2)2 − λv(σ 3 + σπππ2)+ const. (9.38)

It describes nucleons of mass mN = g(|µ2/λ|)1/2 = −gv, a σ meson of massmσ = 2(−µ2/2)1/2 = (2v2λ)1/2 and the isospin triplet of massless pions. Inaddition we observe that in the tree approximation, from (9.29)

〈0|Aaµ(x)|πb(q)〉 = 〈0|σ ′|0〉〈0|∂µπa(x)|πb(q)〉

= −〈0|σ ′|0〉iqµδab exp(−iqx)

In the tree approximation the other terms in the axial current (9.29) give a vanishingcontribution to this matrix element. Comparing with (9.19) one gets

fπ = −〈0|σ ′|0〉The v or fπ is the only mass scale of the model.

Our last remark in this section is as follows: the model can be extended toaccount for an explicit breakdown of the chiral symmetry. To do this in an

9.4 Goldstone’s theorem 327

isotopically invariant manner one adds to L a term −εσ ′. Then minimizingthe potential (9.33) plus the extra term one finds instead of (9.36) the followingequation for the vacuum expectation value v:

−ε − λv3 − µ2v = 0

With the shift (9.37) one now gets

m2π = µ2 + λv2 = −ε/v = ε/ fπ

The axial current is no longer conserved

∂µAaµ(x) = επa(x) (9.39)

and finally we get

∂µAaµ(x) = m2

π fππa(x) (9.40)

which is the operator PCAC relation (9.25) introduced earlier. Another effect ofthe εσ ′ term is that it defines the unique physical vacuum. This vacuum alignmentproblem will be discussed in more detail later on.

9.4 Goldstone bosons as eigenvectors of the mass matrix and poles of Green’sfunctions in theories with elementary scalars

We have learned in the previous section that spontaneous breakdown of thechiral symmetry in the σ -model gives massless Goldstone bosons in the physicalspectrum. Here we study the problem of Goldstone bosons in theories withelementary scalars (QCD-like theories are discussed in Section 9.6) in a moregeneral context.

Goldstone bosons as eigenvectors of the mass matrix

First we generalize the discussion of Section 9.3 and show in the tree approxi-mation that in a theory with spontaneously broken continuous global symmetrythe mass matrix of the scalar sector has eigenvalues zero. The correspondingeigenvectors are Goldstone boson fields. Next we give the general proof ofGoldstone’s theorem (Goldstone, Salam & Weinberg 1962) which does not relyon perturbation theory. This proof can also be easily extended to theories withoutelementary scalars.

Consider a theory described by a lagrangian L which has some global continuoussymmetry G generated by charges Qa . Among other things, the theory containselementary scalar fields �i , i = 1, . . . , n, which we take as real; any complex


representation can always be turned into a real one by doubling the number of thebasis vectors. The piece Ls of the lagrangian containing the scalar fields �i is

Ls = 12∂µ�i∂

µ�i − V (�i )+ · · · (9.41)

where the unspecified terms, irrelevant for our discussion in the tree approximation,are possible couplings of the scalar fields to other fields of the theory. The V (�i )

is a real polynomial in fields �i , of the fourth order at most, if the theory is tobe renormalizable. We assume furthermore that the set �i forms a multiplet �

transforming under some irreducible representation of the group G of the globalcontinuous symmetry of the lagrangian L (if the original fields form a reduciblemultiplet we change the basis to write it as a direct sum of irreducible multiplets)

�i → �′i ≈ �i − i�aT a

i j�i (9.42)

Matrices T a form the representation of charges Qa in the n-dimensional spaceof scalar fields �i . Because iT a must be real, T a is purely imaginary and beinghermitean it is antisymmetric. The transformation rule (9.42) is equivalent to thefollowing commutation relation between charges Qa and fields �i (see (9.32)):

[Qa(t),�i (x, t)] = −T ai j� j (x, t) (9.43)

We are interested in theories in which the global symmetry G is spontaneouslybroken by the vacuum which is such that some of the fields �i have non-vanishingvacuum expectation values

〈0|�i |0〉 = vi �= 0 (9.44)

In the tree or classical approximation, when the energy density of the vacuum stateis just given by the minimum of the potential V (�i ), one then concludes that forthe spontaneous breakdown of the symmetry G the condition for the minimum ofV (�i )

0 = δV (�i ) = ∂V

∂�iδ�i = −i

∂V

∂�i�aT a

i j� j (9.45)

must have solutions with �i = vi �= 0 for some of the �i s. Notice that if the vectorv = (v1 . . . vn)

T is a solution of (9.45) then it follows from the G invariance of thepotential V that the vector

v′ = exp(−i�aT a)v (9.46)

is another solution to this equation. Thus, for a given potential V which gives thespontaneous breakdown of the symmetry G the vacuum is infinitely degenerateand any choice of it is physically equivalent (a choice of a solution v is a choiceof the vacuum). Of course, different potentials may, in principle, give physically


different, non-degenerate, vacua. Possible patterns of spontaneous symmetrybreaking are discussed in more detail in the next section.

Choosing some solution v of (9.45) as the physical vacuum of our theory it isconvenient to define the physical fields �′

�′i = �i − vi (9.47)

and to rewrite in terms of them the potential V

V (�i ) = const + 12 M2

ik(�− v)i (�− v)k + · · · (9.48)

where

M2ik =

(∂2V

∂�i∂�k

)�=v

(9.49)

is the mass matrix for the physical scalar fields. The term linear in fields does notappear in (9.48) because of (9.45). Differentiating (9.45) one gets the followingresult:

M2ik iT a

i jv j = 0 (9.50)

which suggests that the mass matrix has eigenvectors T av corresponding to zeroeigenvalues. To discuss this point more precisely we divide the generators of G intotwo groups. The first one consists of those which remain unbroken with our choiceof the vacuum. Denoting their matrix representation by Y i , i = 1, . . . , n, one hasY Iv = 0; the vacuum is invariant with respect to Y i and (9.50) does not give anynew information. To the second group belong generators which are spontaneouslybroken. Denoting their matrix representation by Xi , i = n + 1, . . . , N (note thevalues of the index i) we have Xiv �= 0 and (9.50) implies the existence, for eachbroken generator of G, of the eigenvectors Xiv corresponding to zero eigenvaluesof the mass matrix. The massless boson fields (normal coordinates) are then thefollowing:

�l = i(Xlv)i�i (9.51)

We recall at this point that our discussion is for spontaneously broken globalsymmetries. For spontaneously broken gauge symmetries the Goldstone bosonfields can be gauged away and do not appear in the physical spectrum. Thesedegrees of freedom are used for longitudinal components of the massive gaugeboson. This so-called Higgs mechanism is discussed in Section 11.1.

We conclude these considerations with several useful remarks. Obviously, ifsome of the fields have non-vanishing vacuum expectation values, some generatorsof G must be broken. The generators which remain unbroken must form the algebra


of some subgroup H of group G (it may, of course, be that H ≡ 1). Indeed, if it isnot the case then

[Y i , Y j ] = ici jkY k + ici jl Xl, i, j, k ≤ n, l > n (9.52)

with some of the ci jl different from zero. Acting with both sides of (9.52) on thevector v we get a contradiction. Therefore

ci jl = 0, i, j ≤ n, l > n (9.53)

and the unbroken generators form the algebra of a group. Furthermore, it is easy tosee that broken generators transform under some representation of the subgroup H

[Y i , X j ] = ici jk Xk, i ≤ n, j, k > n (9.54)

This follows from the property (9.53) and from the antisymmetry of the structureconstants in indices i, j, k. Thus, the r.h.s. of (9.54) cannot contain a term ci jlY l ,where i, l ≤ n, j > n. Eqs. (9.54) and (9.51) taken together tell us that Goldstonebosons transform under some representation of the unbroken subgroup H . Finally,in many physical problems one can define a parity operation P which leaves theLie algebra of the group G invariant and such that

PY i P−1 = Y i , P Xi P−1 = −Xi (9.55)

This is the case, for example, for chiral symmetry; see Section 9.1. Then there isone more useful and obvious relation

[Xi , X j ] = ici jkY k, i, j > n, k ≤ n (9.56)

General proof of Goldstone’s theorem

We turn now to the general proof of Goldstone’s theorem (Goldstone, Salam &Weinberg 1962) in the class of theories specified in the beginning of this section.One can show that spontaneous breakdown of some global continuous symmetryG of the lagrangian implies poles at p2 = 0 in certain Green’s functions andconsequently implies the existence of massless bosons in the physical spectrum.

Let us consider the Green’s function

Gaµ.k(x − y) = 〈0|T ja

µ(x)�k(y)|0〉 (9.57)

where j aµ(x) is the current corresponding to a generator Qa of the symmetry G and

�k(y) belongs to an irreducible multiplet of real scalar fields. The Green’s functionGa

µ,k(x − y) satisfies a Ward identity obtained by differentiating (9.57) with properaccount of the �-functions involved in the definition of the T -product

∂µ

(x)Gaµ,k(x − y) = δ(x0 − y0)〈0|[ j a

0 (x),�k(y)]|0〉 (9.58)


This is an example of the so-called non-anomalous Ward identity (see Sec-tion 10.1); for the discussion of anomalies in theories with fermions see Chapter 13.Using the relation

[ j a0 (x, t),�k(y, t)] = −T a

k j� j (y, t)δ(x − y) (9.59)

which follows from the assumed transformation properties, see (9.43), of fields �i

under the group G, one gets the following Ward identity (translational invarianceis assumed)

∂µ

(x)Gaµ,k(x − y) = −∂(x − y)T a

k j 〈0|� j (0)|0〉 (9.60)

A few words are worthwhile here about renormalization properties of (9.60) (seealso the next chapter). The Ward identity is, in principle, derived for bare currentsand fields for which transformation properties (9.59), are assumed. However, theconserved currents are not subject to renormalization and the field renormalizationconstants �B

i = Z 1/2i �R

i = Z 1/2�Ri do not carry the group index because the

lagrangian is G invariant. Thus, the renormalized fields transform under the groupG in the same way as the bare fields and the same Ward identity (9.60) holdsboth for bare and renormalized Green’s functions. Let us take it as a relation forrenormalized quantities. Introducing the Fourier transform Ga

µ,k(p)

Gaµ,k(x − y) =

∫d4 p

(2π)4exp[−ip(x − y)]Ga

µ,k(p) (9.61)

one gets from (9.60) the following:

ipµGaµ,k(p) = T a

k j 〈0|� j (0)|0〉 (9.62)

This last equation implies that if some of the vacuum expectation values 〈0|� j |0〉do not vanish then there are poles at p2 = 0 in the Green’s functions Ga

µ,k(p),related to these 〈0|� j |0〉 by (9.62). Indeed, from Lorentz invariance the generalstructure of Ga

µ,k(p) can be written as

Gaµ,k(p) = pµFa

k (p2) (9.63)

and therefore

Fak (p2) = −iT a

k j 〈0|� j (0)|0〉(1/p2) (9.64)

Our last step is to show that poles at p2 = 0 imply the existence of massless bosonsin the physical spectrum. To this end we consider the matrix element

〈0| j aµ(x)|π k(p)〉 = i f a

k pµ exp(−ipx) (9.65)

where the vector |π k(p)〉 describes a particle of mass mk which is a quantum of the


field �k . Using the reduction formula (see Section 2.7) one can relate the matrixelement 〈0| j a

µ(x)|π k(p)〉 to the Green’s function Gaµ,k(p)

〈0| j aµ(x)|π k(p)〉 =

∫d4 y d4z f p(z)G

aµ,k′(x − y)iG−1

k′k(y − z) (9.66)

where

f p(z) = exp(−ipz)

and

Gk′k(y − z) =∫

d4q

(2π)4

−δk′k

q2 − m2k + iε

exp[−iq(y − z)]∫d4 y G−1(x − y)G(y − z) = δ(x − z)

(9.67)

Writing exp(−ipz) as exp(−ipx) exp[ip(x− y)] exp[ip(y−z)] we get from (9.66)the following result:

〈0| j aµ(x)|π k(p)〉 = lim

p2→m2k

exp(−ipx)Gaµ,k′(p)iG−1

kk (p)

= limp2→m2

k

−i exp(−ipx)Gaµ,k(p)(p2 − m2

k) (9.68)

Comparison with (9.65) gives us our final equation:

limp2→m2

k

Gaµ,k(p)(p2 − m2

k) = − f ak pµ (9.69)

If Gaµ,k(p) is one of those Green’s functions which (according to (9.64)) have for

spontaneously broken symmetry a pole at p2 = 0 then (9.69) implies m2k = 0 and

f ak �= 0:

f ak = iT a

k j 〈0|� j (0)|0〉 (9.70)

Thus there must be massless bosons |�a(p)〉 = iT ak j 〈0|� j (0)|0〉|π k(p)〉 in the

physical spectrum corresponding to each broken generator which does not leavethe vacuum invariant T a

k j 〈0|� j (0)|0〉 �= 0. If the broken generators and the cor-responding currents form a single real irreducible representation of the unbrokensubgroup H which leaves the vacuum invariant then from Schur’s lemma

〈0| j aµ|π k(p)〉 ∼ δak (9.71)

and

f ak = f aδak

9.5 Patterns of spontaneous symmetry breaking 333

For instance, for the σ -model with 〈0|σ ′|0〉 �= 0 we get from (9.70), with matricesT a

k j from Appendix E,

fπ = −〈0|σ ′|0〉 (9.72)

as the generalization of the result obtained in the tree approximation.Goldstone’s theorem does not hold for spontaneously broken gauge symmetries

(see Section 11.1). The breakdown of the presented proof is most easily seen if wework in a ‘physical’ gauge in which the gauge field has only physical degrees offreedom. Gauge fixing then requires a specification of some four-vector nµ and thegeneral form of the matrix element involving the gauge source current is no longergiven by (9.63): a term proportional to nµ can be present. For a discussion in acovariant gauge see, for example, Bernstein (1974).

9.5 Patterns of spontaneous symmetry breaking

For a theory with some global symmetry G and a given multiplet content of fields,patterns of spontaneous symmetry breaking are determined by the properties of thevacuum defined by the non-vanishing of the vacuum expectation values of somefields. In theories with elementary scalars the non-vanishing vacuum expectationvalues may already appear in the tree approximation as a solution minimizingthe assumed potential V (�) or may be generated by radiative corrections (Sec-tion 11.2). In theories without elementary scalars the symmetry is spontaneouslybroken by some condensates (for example, 〈0|��|0〉) which, in the absence ofreliable methods for calculating non-perturbative effects, are usually introducedby assumption. A choice of the vacuum specifies the pattern of the spontaneoussymmetry breaking which may, in general, be such that some subgroup H of Gremains unbroken. As we have already mentioned in the previous section for anychoice of the vacuum specified by the non-vanishing vacuum expectation valuesthere exists a set of physically equivalent vacua, which also minimize the samepotential V (�), obtained from the original one by the set of unitary transformationsbelonging to G. Different choices of the physical vacuum among the degeneratevacua give the same pattern of symmetry breaking; they simply correspond todifferent orientations of the unbroken subgroup H in the group G which can betransformed one into the other by a redefinition of the basis.

The question we would like to discuss here is what are the possible patternsof spontaneous breaking of a given symmetry G. Although different patterns, ifthey exist, must, in general, correspond to different potentials V (�), no explicitreference to the potential is necessary for our discussion.

Our first example is a group of orthogonal transformations, O(3) for definite-ness, and the scalar fields are assumed to form a real multiplet � = (�1,�2,�3)

T


transforming under the vector representation. The matrix representation of gener-ators Qi in the space of �s is

Tx = 0 0 0

0 0 −i0 i 0

Ty = 0 0 i

0 0 0−i 0 0

Tz = 0 −i 0

i 0 00 0 0

(9.73)

A possible choice for the vacuum expectation value of the multiplet � is, forinstance,

〈0|�|0〉 = 0

0v

≡ �0

The vacuum defined this way is invariant under the transformations generated byQz: taking the transformed vacuum exp(−iαQz)|0〉 and using (9.32) we get

〈O| exp(iαQz)� exp(−iαQz)|0〉 ∼= �0 + iα〈0|[Qz,�]|0〉 = �0 − iαTz�0 = �0

(9.74)However, the vacuum defined by (9.74) breaks spontaneously the other twogenerators:

iTx�0 = 0

v

0

, iTy�0 =−v

00

(9.75)

Thus the group O(3) is broken to the group O(2) of rotations around the z axis.There are two Goldstone bosons corresponding to two linearly independent vectorsiTx�0 and iTy�0. Any other choice of the vacuum expectation value

〈0|�|0〉 = v1

v2

v3

(9.76)

which can be obtained from the original one by unitary transformations belongingto G gives the same pattern of symmetry breaking but this time with the combi-nation v1Tx + v2Ty + v3Tz unbroken. Both theories are physically equivalent andtransform one into the other by a redefinition of the basis. This would not be thecase if there was, in space, a physically distinguished direction, that is if the O(3)symmetry was also broken explicitly. This point will be discussed in detail in thenext chapter.

Consider now unitary groups with scalars in a real representation. Any complexfield � can be written in a real basis by defining two real fields

ϕi = (1/√

2)(�i +�∗i )

χi = (1/√

2i)(�i −�∗i )

}(9.77)

9.5 Patterns of spontaneous symmetry breaking 335

For the group U (1) of the unitary transformations

� → �′ = exp(−i�)�

the real representation is

�r → �′r = exp(−i�T )�r

where

�r =(

ϕ

χ

)(9.78)

and T is purely imaginary

T =(

0 i−i 0

)because �r must be real. Obviously, in this case, there is just one pattern of spon-taneous symmetry breaking and any non-zero choice of the vacuum expectationvalue 〈0|�r|0〉 = (v1v2)

T is equally good.Our next example is the group SU (2) with a doublet of complex fields �1 and

�2. In the real representation

�r =

ϕ1

χ1

ϕ2

χ2

the generators of the SU (2) transformations are

T 1 =

i

−ii

−i

, T 2 =

0−i

−ii

i0

,

T 3 =

i

−i0

0−i

i

(9.79)

Let us take, for instance, 〈0|�r|0〉 = (0, 0, v, 0)T. Following the discussion ofour first example we see that the group is broken completely because the threevectors iT a〈0|�r|0〉 �= 0 are linearly independent. All other possible non-vanishingvacuum expectation values can be obtained from the one above by unitary SU (2)transformations and, thus, give the same pattern of symmetry breaking.


It is easy to extend the analysis to the SU (2) × U (1) group with one complexdoublet of scalar fields (Glashow–Salam–Weinberg model). In this case we haveone more generator

T =

i

−i0

0i

−i

but the vector iT 〈0|�r|0〉 is not linearly independent from the other three vectorsiT a〈0|�r|0〉. For instance, for 〈0|�r|0〉 = (0, 0, v, 0)T we have T 〈0|�r|0〉 =−T 3〈0|�r|0〉 and the combination T + T 3 remains unbroken. Other choices ofthe vacuum give only different unbroken combinations of the generators T and T a

but a redefinition of the basis transforms one case into another. Thus, again onlyone pattern of symmetry breaking is possible.

The spontaneous breaking of chiral symmetry in the σ -model can be discussed ina similar manner. Using the real basis �r = (π1π2π3σ

′)T and the matrices T a and

5T a collected in Appendix E we see that, for example, for 〈0|�r|0〉 = (0, 0, 0, v)T

the generators 5T a are broken and the T as remain unbroken: SUL(2)× SUR(2) →SUL+R(2). This corresponds to the standard choice 〈0|σ ′|0〉 �= 0 and to the πsforming a triplet under the SU (2). However, any other choice of vacuum amongthe SU (2) × SU (2) degenerate set gives the same pattern of symmetry breaking.Take, for example, 〈0|�r|0〉 = (v, 0, 0, 0)T. Then the generators 5T 2, 5T 3and T 1

remain unbroken and we see from the commutation relations (9.9) that they alsoform an SU (2) algebra. We can call this group the isospin group and interpretas physical pions those combinations of the fields π i and σ which transform as atriplet under this new SU (2), thus obtaining the same theory as before.

So far we have been discussing examples with only one possible pattern ofspontaneous symmetry breaking. This is, of course, not always the case. As ourlast example we consider the group SU (3) with a real multiplet � of scalar fieldstransforming under the adjoint representation of the SU (3)

[Qa,�b] = −T abc�c

where

T abc ≡ −icabc (9.80)

It is convenient to write the scalar multiplet in the matrix form

� = 12�

bλb, b = 1, . . . , 8 (9.81)

9.6 Goldstone bosons in QCD 337

and λb are the Gell-Mann matrices. The matrix � is hermitean and �i i = 0 so ithas eight independent elements. The transformation rule for the matrix � is

� → �′ = exp(−i�a 12λ

a)� exp(i�a 12λ

a) ∼= �− i�a[ 12λ

a, �] (9.82)

and because of (9.81) it is equivalent to

� → �′ = exp(−i�aT a)�

with T a given by (9.80), for the multiplet �. Because the matrix � is hermitean itcan be diagonalized by unitary transformations and written in terms of commutinggenerators of SU (3). Taking them as λ3 and λ8 we can write the most general formof the vacuum expectation values of the scalar fields as follows:

〈0|�|0〉 = 12v3λ

3 + 12v8λ

8 (9.83)

First of all we notice that in our example the maximal possible breaking isSU (3) → U3(1) × U8(1) because it is clear from (9.83) and (9.82) that λ3

and λ8 remain unbroken for any choice of v3 and v8. In this case we havesix Goldstone bosons. However, another pattern of symmetry breaking is alsopossible, namely SU (3) → SU (2) × U (1). If we assume v3 = 0, v8 �= 0 thegenerators λ1, λ2, λ3 (SU (2)) and λ8(U (1)) remain unbroken; if v3 = +√3v8

the λ6, λ7, 12(λ

3 − √3λ8)(SU (2)) and 1

2(√

3λ3 + λ8) (U (1)) are unbroken; ifv3 = −√3v8 the λ4, λ5, 1

2(λ3 + √

3λ8) (SU (2)) and 12(√

3λ3 − λ8) (U (1))are unbroken. It is also worth noticing that a vacuum expectation value givingSU (3) → SU (2) × U (1) cannot be transformed by unitary transformationsbelonging to SU (3) to one which gives SU (3) → U (1) × U (1) and they mustcorrespond to different potentials V (�). However, for each pattern there is aninfinitely degenerate set of possible vacua obtained from (9.83) (with specified v3

and v8) by unitary SU (3) transformations.For a given pattern of spontaneous symmetry breaking one usually works from

the beginning in a convenient basis with a specified physical interpretation of thegroup generators and of the fields, as we do for the σ -model in Section 9.3 and forQCD in Section 9.6.

9.6 Goldstone bosons in QCD

The purpose of this section is to extend the proof of Goldstone’s theorem to theorieslike QCD, with no elementary scalars. Chiral symmetry of the massless QCDlagrangian has been discussed in Section 9.1. Considerable phenomenologicalevidence has then been summarized that the group SU (2) × SU (2) is an approx-imate symmetry of the strong interactions which is spontaneously broken by thevacuum. One should be aware, however, of the fact, that our understanding of


the underlying mechanism of spontaneous symmetry breaking in theories withoutelementary scalars is not yet satisfactory. It has at least been realized thatonly asymptotically free theories allow spontaneous chiral symmetry breakingat reasonable momentum scales (Lane 1974, Gross & Neveu 1974). It is notour aim here to review partial results suggesting that this indeed happens; weshall assume the spontaneous breakdown of chiral symmetry and then show thatit implies the existence of massless bosons (composite states) in the physicalspectrum. It is nevertheless useful to give at least an intuitive argument in favourof the spontaneous breaking of chiral symmetries in asymptotically free theories.This relies on the fact that the gauge coupling in QCD becomes strong at largedistances. Assuming that it becomes arbitrarily strong we expect that the groundstate of the theory has an indefinite number of massless fermion pairs which canbe created and annihilated by the strong coupling. We still expect the ground stateto be invariant under Lorentz transformations so these pairs must have zero totalmomentum and angular momentum. Thus we find that the vacuum |0〉 has theproperty that operators which destroy or create such a fermion pair have non-zero vacuum expectation values 〈0|�R�L|0〉 �= 0, 〈0|�L�R|0〉 �= 0; we omitindices here. In the following we assume that the chiral symmetry of QCD isindeed spontaneously broken by non-vanishing vacuum expectation values of somefermion condensates.

We consider QCD with two flavours and the chiral symmetry SUL(2)× SUR(2).Let us introduce scalar operators � and �†

�i j = �L j�Ri , (�†)i j = �R j�Li (9.84)

(i, j = 1, 2 are flavour indices). From the transformation properties of the fields�Li and �R j under the chiral group

� ′Ri = (UR)ik�Rk

�′†L j = �

†Lk(U

†L)k j

it follows that

�′ = UR�U †L

(�†)′ = UL�†U †

R

}(9.85)

We assume

〈0|�i j |0〉 �= 0, 〈0|�†i j |0〉 �= 0

and in addition

〈0|�i j |0〉 = 〈0|�†i j |0〉


The latter equality ensures that parity is not spontaneously broken since then

〈0|�iγ5� j |0〉 = 〈0|(�−�†) j i |0〉 = 0 (9.86)

(we always work in a specified basis of chiral fields).The vacuum expectation value 〈0|�|0〉, being a hermitean 2 × 2 matrix, can be

most generally written as (see the discussion preceding (9.83))

〈0|�|0〉 = v1l = v3σ3 (9.87)

where σ 3 is the diagonal Pauli matrix. We want the chiral SUL(2) × SUR(2)symmetry to be broken to the SUL+R(2) symmetry which is the isospin symmetryin our basis. Therefore we assume v �= 0, v3 = 0 in (9.87): it is clear from (9.85)that the vacuum expectation value v1l is invariant under the simultaneous UL = UR

transformation. Notice the difference with the σ -model with one complex doublet,or four real fields, where the SU (2)× SU (2) → SU (2) is the only possible patternof symmetry breaking; now the 〈0|�|0〉 is a 2 × 2 complex hermitean matrix. Wecan also write

〈0|�i j |0〉 = 12〈0|��|0〉δi j (9.87a)

where

�� = �1�1 + �2�2

We are ready to prove Goldstone’s theorem. The proof is similar to that inSection 9.4 and it is based on Ward identities for certain appropriate Green’sfunctions. First we collect the useful equal-time commutation relations followingfrom (9.59) and (9.85)

[ j a0L(x, t),�(y, t)] = �(x, t)T aδ(x − y)

[ j a0R(x, t),�(y, t)] = −T a�(x, t)δ(x − y)

[ j a0L(x, t),�†(y, t)] = −T a�†(x, t)δ(x − y)

[ j a0R(x, t),�†(y, t)] = �†(x, t)T aδ(x − y)

(9.88)

where the j a0L ( j a

0R) are the zeroth components of the currents of the SUL(2)(SUR(2)) transformations and the matrices T a represent the generators of theSUL(2) (when to the right of the operator �) and of the SUR(2) (when to theleft of the �) in the space of chiral doublets �L and �R, respectively. The abovecommutation relations are valid both for the bare and the renormalized quantitiesbecause chiral symmetry is an exact symmetry of the lagrangian (see also the nextchapter). From now on we always refer to renormalized quantities. For the proofof Goldstone’s theorem we use a Ward identity for Green’s functions involvingcomposite operator �a(x) defined as follows:

�a(x) = Tr[�(x) · T a]


where

�(x) = i(�−�†), �† = � (9.89)

and therefore

�a(x) = i(�i j −�†i j )T

aji = i�T aγ 5�

(T a = 12σ

a, σ a are Pauli matrices). Using the commutation relations (9.88) we getthe following result for the axial currents:

[Aa0(x, t),�(y, t)] = −i{T a,�(y, t)+�†(y, t)}δ(x − y) (9.90)

where the curly bracket on the r.h.s. means anticommutator. Using this last resultwe finally arrive at the desired Ward identity:

∂µ

(x)〈0|TAaµ(x)�

b(y)|0〉 = −iδ(x− y)〈0|Tr[{T a,�(y)+�†(y)}T b]|0〉 (9.91)

(it is unaltered by the chiral anomalies discussed in Chapter 13). UsingTr[T aT b] = 1

2δab and (9.87) one gets

∂µ

(x)〈0|TAaµ(x)�

b(y)|0〉 = −2viδ(x − y)δab (9.92)

This result is analogous to (9.60) and it implies a pole at p2 = 0 in the Fouriertransform Gab

µ,�(p) of the Green’s function Gabµ,�(x − y)

Gabµ,�(x − y) = 〈0|TAa

µ(x)�b(y)|0〉 =

∫d4 p

(2π)4exp[−ip(x − y)]Gab

µ,�(p)

(9.93)

Following the steps (9.62)–(9.64) we get

Gabµ,�(p) = pµ

2v

p2δab = pµ

〈��〉p2

δab (9.94)

The pole at p2 = 0 in the Green’s function Gabµ,�(p) implies the existence in the

physical spectrum of the massless Goldstone boson with the quantum numbers ofthe operators �a . The proof goes as in Section 9.4. The only modification is thatsome care must be taken about the interpretation of the operator �(x). Indeed,consider the Green’s function

Gab(x − y) = 〈0|T�a(x)�b(y)|0〉 (9.95)

A single particle state with appropriate quantum numbers gives a pole at p2 = m2

in the Fourier transform of (9.95)∫d4x Gab(x) exp(ipx) ≈

p2→m2

iZ2

p2 − m2δab (9.96)


where Z is a finite dimensionful constant (�a is a renormalized operator). It doesnot carry the group index because G is an exact symmetry of the lagrangian.

Introducing the physical, in the sense of Chapter 2, field πa(x):

πa(z) = Z−1�a(x) (9.97)

we repeat the step (9.65)

〈0|Aaµ(x)|πb(p)〉 = i fπδ

ab pµ exp(−ipx) (9.98)

and the steps (9.66)–(9.68)

〈0|Aaµ(x)|πb(p)〉 ≈

p2→m2−i exp(−ipx)Z−1Gab

µ,�(p)(p2 − m2) (9.99)

to get

limp2→m2

Gabµ,�(p)(p2 − m2) = −Z fπδ

ab pµ (9.100)

Since Gabµ,�(p) has the pole (9.94) due to the spontaneous breakdown of chiral

symmetry, (9.100) implies m2 = 0 and

Z fπ = −〈��〉 (9.101)

This completes the proof of Goldstone’s theorem. In this case we have two lowenergy parameters: fπ and 〈��〉.

10

Spontaneous and explicit global symmetry breaking

So far we have been discussing theories described by lagrangians having someexact global symmetry G spontaneously broken by the vacuum. Now we wouldlike to introduce the possibility that the symmetry G of the lagrangian is alsobroken explicitly by a small perturbation. Several interesting physical problemsfall into this category. We begin with a systematic discussion of Ward identitiesbetween Green’s functions involving global symmetry currents. This will also beuseful as a supplement to our use of Ward identities in the earlier chapters and asan introduction to Chapter 13 devoted to the problem of anomalies.

10.1 Internal symmetries and Ward identities

Preliminaries

We are interested in theories described by lagrangians L such that

L = L0 + L1

(or H = H0 + H′), where L0 is invariant under global internal symmetry G andL1 explicitly breaks this symmetry. In addition the symmetry G may or may notbe spontaneously broken. We assume that L1 can be treated as a perturbationwhose effect can be calculated as a perturbative expansion in an appropriate smallparameter. Furthermore, let the bare fields in the lagrangian L form a basis forsome definite representation R of the symmetry group G of the lagrangian L0

δ�i = −i�aT ai j� j ≡ �aδa�i (10.1)

(� stands for boson or fermion fields).

342

10.1 Internal symmetries and Ward identities 343

The bare currents j aµ(x) constructed according to Noether’s theorem

j aµ(x) = −i

∂L∂(∂µ�i )

T ai j� j (10.2)

which are conserved currents of the G symmetric classical theory described by L0,remain unchanged when L0 → L if L1 does not contain derivatives of the fieldspresent in the j a

µ(x); we assume this to be the case.Under an infinitesimal global symmetry transformation

�′i (x) = �i (x)+�aδa�i (x) (10.3)

the lagrangian L transforms as follows:

L′(x) = L(x)+�aδaL(x) = L(x)+�aδaL1(x) (10.4)

and, using the classical equations of motion,

∂µ j aµ(x) = δaL1(x) (10.5)

In the following we shall also need the variation of the same lagrangian L underlocal transformations

�′i (x) = �i (x)+�a(x)δa�i (x) (10.6)

Remembering that L0 is invariant under global transformations (10.3) and that L1

does not contain derivatives of the fields we get

δL(x) = �a(x)δaL(x)+ j aµ(x)∂

µ�a(x) (10.7)

We want to derive Ward identities for bare regularized or for renormalizedGreen’s functions following from the symmetries of the lagrangian. We assumein this section that the UV regularization of the theory does preserve its classicalsymmetries. As we will see in Chapter 13 this is not true in theories with chiralfermions. In such theories some of the Ward identities derived here are afflictedwith anomalies.

Let us first consider the case in which the symmetry G is an exact symmetry ofthe lagrangian, i.e. L1 = 0. If the regularization preserves this symmetry then thenecessary counterterms are also G-invariant. Both bare regularized and renormal-ized Green’s functions are G symmetric. The renormalized fields transform underthe same representation R of G as the bare fields: the renormalization constants areG-invariant. The same discussion applies when the symmetry G is spontaneouslybroken since in this case also only symmetric counterterms are needed: this followsfrom (2.161). Whether G is spontaneously broken or not we are able to derive thesame Ward identities.

344 10 Spontaneous and explicit global symmetry breaking

With L1 present we consider the case in which symmetry breaking operatorshave dimension d < 4 (L1 then involves dimensionful constants). This is so-calledsoft explicit symmetry breaking. Then the counterterms required by L1, beingof dimension lower than or equal to d (see the discussion in Section 4.2) donot affect counterterms of dimension 4. Thus, for such a ‘soft’ breaking thewave-function renormalization constants, included in counterterms of dimension4, remain symmetric under G (strictly speaking there exists a renormalizationscheme such that they remain symmetric) and the renormalized fields of thetheory with softly broken symmetry G transform under G in the same wayas the bare fields. Our general remarks can be illustrated with the minimalsubtraction renormalization scheme used for QCD with the mass term m��; thewave-function renormalization constants are mass-independent and thus the sameas in the chirally symmetric theory.

Ward identities from the path integral

Given all these preliminary constraints we can proceed to derive Ward identities us-ing the functional integral formulation of quantum theory. Consider the generatingfunctional W [J ], for example, for bare regularized Green’s functions

W [J ] = N∫

D�i exp

(i

{∫d4x [L(�i )+ J i�i ]

})(10.8)

The integral is invariant under the change of integration variables defined by (10.6).If the integration measure is invariant,† using (10.7) we get

0 =∫

d4x∫

D�i exp

(iS + i

∫d4x J i�i

)× [�a(x)δaL+ j a

µ(x)∂µ�a(x)+ J i (x)�a(x)δa�i (x)] (10.9)

The crucial point in deriving (10.9) is the invariance of the integration measureunder the transformation (10.6). It breaks down for chiral transformations intheories with chiral fermions (see Chapter 13). In other words, in this case thereis no regularization of the path integral and no renormalization prescription forthe Green’s functions which preserve the full chiral symmetry and anomalies arepresent. We now define the Green’s functions

〈0|T A(x)∏

i

�i (yi )|0〉 ∼∫

D�i A(x)∏

i

�i (yi ) exp(iS) (10.10)

where A(x) denotes one of the j aµ(x), δ

aL(x) and δa�i (x). One should rememberthat the quantity on the l.h.s. of (10.10) is just a short-hand notation for its r.h.s.

† We consider cases in which det(D�′i /D� j ) = 1 + O(�2). This is true when generators T a in (10.1) are

traceless. We also assume the absence of anomalies.


The notation has been chosen, however, to make easy contact with the operatorlanguage. Integrating (10.9) by parts (remember the remarks following (2.55))differentiating it functionally with respect to the sources Ji and then setting themto zero we get the following general Ward identity:

(∂/∂xµ)〈0|T jaµ(x)

∏i

�i (yi )|0〉 = 〈0|T δaL(x)∏

i

�i (yi )|0〉

− i∑

i

δ(x − yi )〈0|T δa�i (yi )∏j �=i

� j (y j )|0〉

(10.11)

Ward identity (10.11) is valid irrespective of whether the symmetry is sponta-neously broken or not. If the symmetry G is an exact global symmetry of thelagrangian the first term on the r.h.s. of (10.11) vanishes. Under the constraint ofexplicit breaking being only soft the Ward identity is valid both for bare regularizedand for the renormalized Green’s functions.

We may also be interested in Ward identities for Green’s functions involvingseveral symmetry currents j a

µ and/or the symmetry breaking operator L1. Wetake the latter to have well-defined transformation properties under the symmetrygroup. The most compact way to derive such Ward identities is based on thegeneral idea (see Chapter 8) of using background fields to get, by construction,an action which is exactly invariant under certain symmetry transformations. Inour case the background fields are the sources j i , Aa

µ and K for the operators �i ,j aµ and L1, respectively. We want to introduce them in such a way that the action

S[�, J, A, K ] is invariant under the local transformation (10.6). This is achievedif

S[�, J, A, K ] = S0[�, A] +∫

d4x (J i�i + K · L1) (10.12)

where S0[�] is the part of the action symmetric under global transformation (10.1)and the background fields Aa

µ have been introduced as gauge fields by changingthe normal derivatives in S0 into covariant derivatives ∂µ → ∂µ + iAa

µT a . Action(10.12) is invariant under (10.6) if simultaneous transformations on A, J and Kdefined as

δAaµ = +∂µ�

a(x)+ cabc�b Ac

δ J i�i = −J iδ�i , δK · L1 = −K · δL1

}(10.13)

The generating functional

W [A, J, K ] = N∫

D�i exp{iS[�, J, A, K ]}


is then invariant under transformations (10.13) of the background fields

W [A, J, K ] = W [A′, J ′, K ′]

The last statement contains all the information equivalent to Ward identitieswhich follow from the global symmetry of the original theory. For infinitesimaltransformations the variation of the functional W [A, J, K ] reads

0 =∫

d4x

[δW

δ Ji (x)δ J i (x)+ δW

δAaµ(x)

δAaµ(x)+

δW

δK (x)δK (x)

](10.14)

or specifying j aµ, for instance, to flavour fermionic currents,

0 =∫

d4x∫D�i exp{iS[A, J, K ,�]}(J iδa�i+K ·δaL1−∂µ jµa+cabc Ab

µ jµc)�a(x)

(10.15)Eq. (10.15) generates the desired Ward identities valid to any order in L1 bydifferentiating with respect to sources J i , K and Aa

µ and setting J i = Aaµ = 0,

K = 1. The method can be generalized to generate Ward identities involving othercomposite operators with well-defined transformation properties under the globalsymmetry group if we introduce sources for such operators.

We stress again that the validity of the symmetry relations for the Green’sfunctions relies on the existence of the regularization and renormalization pro-cedures which preserve this symmetry. In particular renormalization of Green’sfunctions involving products of composite operators, in addition to multiplicativerenormalization, may also require subtractions in the form of polynomials inmomenta in momentum space or of δ-functions and their derivatives in positionspace.

Finally we observe that in the special cases of no massless Goldstone bosonscoupled to the current jαµ , Ward identities like (10.11) can be integrated over space-time and the surface terms neglected. Thus in the case of exact symmetry which isnot spontaneously broken one gets

δa〈0|T∏

i

�i (yi )|0〉 = 0 (10.16)

which simply means an exact invariance of the Green’s functions under the sym-metry transformations. When the symmetry is explicitly broken by perturbation L1

we have

δa〈0|T∏

i

�i (yi )|0〉 = −i∫

d4x 〈0|T δaL∏

i

�i (yi )|0〉

≡ −i〈0|T δa S∏

i

�i (yi )|0〉 (10.17)

Note that in the latter case the symmetry may also be spontaneously broken since


in the presence of perturbation L1 the massless Goldstone bosons acquire masses.This can, for instance, already be seen at the level of the tree approximation, i.e. byanalysing the classical lagrangian.

Comparison with the operator language

The equal-time bare current commutation relations following from the equal-timecanonical commutation relations for the bare field operators are

[ j a0 (x, t), j b

0 (y, t)] = icabc j c0 (x, t)δ(x − y) (10.18)

Relations (10.18) are true in the symmetric theory and also with the symmetrybreaking term L1 included as long as such terms do not contain derivatives of thefields. However, in the presence of the symmetry breaking term L1 the currents areno longer conserved and the corresponding charges are time-dependent. Defining

Qa(t) =∫

d3x ja0 (x, t)

we get from (10.18)

[Qa(t), Qb(t)] = icabc Qc(t) (10.19)

In accordance with the transformation properties (10.1) we have furthermore

[Qa(t),�i (x, t)] = −T ai j� j (x, t) (10.20)

or

[ j a0 (x, t),�i (y, t)] = −T a

i j� j (y, t)δ(x − y) (10.21)

If the symmetry breaking is soft (10.18)–(10.21) hold for the renormalized quanti-ties as well because wave-function renormalization constants can remain invariant(for properly chosen renormalization schemes) and as will be seen shortly thecurrents are not subject to renormalization. The time dependence of the chargesis given by

dQa

dt=∫

d3xd

dtj a0 (x, t) = i

∫d3x [H′(x, t), Qa(t)] (10.22)

where H′ is the symmetry breaking part of the hamiltonian, and therefore

∂µ j aµ(x, t) = i[H′(x, t), Qa(t)] (10.23)

(the surface terms at infinity can be neglected because due to perturbation L1 thewould-be Goldstone bosons acquire masses).


The Ward identity (10.11) is the same as one derived for the time-orderedproduct of operators defined as

T jaµ(x)�(y) = j a

µ(x)�(y)�(x0 − y0)+�(y) j aµ(x)�(y0 − x0) (10.24)

(with a straightforward generalization to products of more than two operators) bydirect differentiation of the Green’s function and use of the commutation relations(10.21). For instance

(∂/∂xµ)〈0|T jaµ(x)�(y)|0〉 = 〈0|T ∂µ j a

µ(x)�(y)|0〉+ δ(x0 − y0)〈0|[ j a

0 (x),�(y)]|0〉 (10.25)

and using (10.21) together with (10.5) for the operator ∂µ j aµ we recover the Ward

identity (10.11). Similarly, one can derive Ward identities for Green’s functionsinvolving products of currents and they are the same as the relations followingfrom (10.13) or (10.15) if we neglect in the current commutators the presence ofthe derivatives of the δ-functions (the so called Schwinger terms). However, inmany cases these terms certainly do not vanish (see Jackiw (1972) for a systematicdiscussion) and one may wonder about the compatibility of the two methods. Theanswer is that the Green’s functions defined by the path integral do not alwayscoincide with the vacuum expectation values of the T -products defined by (10.24).The former are always understood to be regularized or renormalized covariantquantities, whereas the latter are not and they may differ by local terms, theso-called seagull terms, which cancel the Schwinger terms. The path integralapproach tells us that it is always possible to arrange such cancellation by a suitablerenormalization prescription if there exists one which preserves the symmetry ofthe classical lagrangian. Otherwise anomalies appear. We shall come back to theseproblems in some detail in the following.

Ward identities and short-distance singularities of the operator products

Consider, for instance, the Ward identity (10.25) with �(x) replaced by anyoperator O(x). This relation is obtained by differentiating the T -product definedby (10.24). However, generally speaking, the meaning of time-ordering as given by(10.24) is clear everywhere except at coinciding points, i.e. at x0 = y0. Thereforethere is an apparent ambiguity

T A(x)B(0) → T A(x)B(0)+ c1δ(x)+ c2∂xδ(x)+ · · · (10.26)

(Lorentz and other symmetry group indices must, of course, be properly taken intoaccount). The question thus arises of the role of such terms which in momentumspace are polynomials in momentum.


To discuss this point we recall first that equations like (10.25) are for distribu-tions, i.e. they hold in the sense that both sides are integrated over

∫d4x f (x) with

f (x) being any ‘smooth’ test function vanishing at xµ → ±∞. In particular, thederivative ∂µT (x) of the distribution T (x) is defined as follows:∫

d4x f (x)∂µT (x)≡df

∫d4x [−∂µ f (x)]T (x) (10.27)

We also define the Fourier transform of a distribution

(2π)−4∫

d4q f (−q)T (q) =∫

d4x f (x)T (x) (10.28)

where f (q) is the Fourier transform of f (x) and use the notation

T (q) =∫

d4x exp(iqx)T (x)

The Fourier transform (∂µT ) of ∂µT (x) is

(2π)−4∫

d4q f (−q)(∂µT ) =∫

d4x f (x)∂µT (x)

= −∫

d4x ∂µ f (x)T (x) (10.29)

i.e.

(∂µT ) = −iqµT (q) (10.30)

In the other notation∫d4x exp(iqx)∂µT (x) = −iqµ

∫d4x exp(iqx)T (x) (10.31)

We expect the meaningfulness of the Ward identity (10.25) to be related to theshort-distance singularity structure of the distributions on both sides of (10.25).

To see this clearly, let us rederive (10.25) paying explicit attention to thesingularities at x ∼ 0. Using the definition (10.27) the l.h.s. of (10.25) can bewritten as

− limε→0

∫d4x ∂µ f (x)〈0|T jµ(x)O(0)|0〉�(|x0| − ε) (10.32)

Performing Wick’s rotation we can work in the Euclidean space, so that the light-cone singularities of the operator product collapse to the origin. Then the integrand


in (10.32) is finite and unambiguous for ε > 0. Integrating by parts we get

− limε→0

(∫ −ε

−∞+∫ ∞

ε

)dx0

∫d3x ∂µ f (x)〈0|T jµO|0〉

= − limε→0

∫d3x f (x)〈0|O(0) j0(x,−ε)− j0(x, ε)O(0)|0〉

+ limε→0

∫d4x f (x)〈0|T ∂µ jµ(x)O(0)|0〉�(|x0| − ε) (10.33)

In the last term the operator ∂µ has been interchanged with T because, afterexcluding x = 0, T jµ(x)O(0) is regular. We have also assumed the absence ofmassless particles; then the surface terms at ±∞ do not contribute.

Similarly, we may consider the Fourier transform∫d4x exp(ipx)∂µ〈0|T jµ(x)O(0)|0〉

= limε→0

∫d4x [−∂µ exp(ipx)]〈0|T jµO|0〉�(|x0| − ε)

= − limε→0

∫d4x exp(ipx)〈0|T j0(x)O(0)|0〉[δ(x0 + ε)− δ(x0 − ε)]

+ limε→0

∫d4x exp(ipx)〈0|T ∂µ jµ(x)O(0)|0〉�(|x0| − ε) (10.34)

If the limit ε → 0 exists, we have

−ipµT (p) = (p)+∫

d3x exp(ipx)〈0|[ j0(x, 0), O(0)]|0〉 (10.35)

where (p) is the Fourier transform of 〈0|T ∂µ jµ(x)O(0)|0〉.Let us discuss the limit ε → 0 in (10.33) and (10.34) using the OPE. One can

immediately convince oneself that the limit exists if

T jµ(x)O(0) ∼ C1µ(x)O1(0)+ less singular terms (10.36)

where C1µ(x) ∼ O(1/x3), and similarly for ∂µ jµ(x). Indeed, with O(1/x3)

singularity the integrals in (10.33) and (10.34) are convergent. From (10.36) wealso get (see Section 7.4)

[ jµ(x, 0), O(0)] = a1 O1(x, 0)δ(x) (10.37)

(no Schwinger terms, a1 finite). Thus, with no additional subtractions the Wardidentity (10.33) or (10.35) with the equal-time commutator given by (10.37) is awell-defined relation.

Often T jµ(x)O(0) is more singular than O(1/x3) (for example T j emµ (x) j em

ν (0)∼ (1/x6) · 1l). Stronger than O(1/x3) singularity in the OPE of the T -product ofoperators translates into terms additional to the δ(x) term with a finite coefficient in


their equal-time commutators (see Section 7.4). These are the so-called Schwingerterms. However, at the same time subtractions at x = 0 are necessary to have awell-defined limit when ε → 0 in the integrals in (10.33) and (10.34). Dependingon the nature of the singularity, without subtractions these integrals may dependon the order of integration over x0 and x (and be then non-covariant) or may bedivergent. Subtractions are of the form

c1δ(x)+ c2∂(x)δ(x)+ · · ·

possibly with divergent ci . For instance, let

T A(x)B(0) ∼(

1

x2 − iε

)2

1l + · · ·

Performing Wick’s rotation and integrating in Euclidean space we conclude thatfor a well-defined ε → 0 limit in the l.h.s. integral in (10.33) a subtraction of theform

iπ2 ln εδ(0)+ cδ(0)

is necessary. The finite constant c is to be fixed by some renormalizationconditions. This subtractive renormalization of the Green’s functions comes inaddition to the multiplicative renormalization of operators. The path integralapproach teaches us that when regularization and renormalization schemes existwhich preserve the classical symmetry of the lagrangian it is possible to definethe renormalized T -products so that they satisfy Ward identities in a form like(10.11), i.e. as if the Schwinger terms were absent. In other words by suitablerenormalization the Schwinger terms can be cancelled by subtraction terms. Ourlast remark is that in applications of Ward identities to physical problems oneshould pay attention to the fact that part of their content may be subtractiondependent.

Renormalization of currents

As the first application of (10.11) we discuss the renormalization properties ofcurrents and their divergences. Let us consider, for instance, the following specificcase of (10.11):

∂µ

(x)〈0|T jaµ(x)�

b(y)�c(z)|0〉 = 〈0|T ∂µ j aµ(x)�

b(y)�c(z)|0〉− δ(x − y)T a

bd〈0|T�d(y)�c(z)|0〉+ δ(x − z)T a

dc〈0|T�b(y)�d(z)|0〉 (10.38)


We take (10.38) to be for bare quantities. In the symmetry limit

∂µ j aµ = 0 (10.39)

�aB = Z 1/2

2 �aR (10.40)

where Z2 is invariant under symmetry transformations. Expressing the barefermion fields in (10.38) by the renormalized fields and dividing both sides byZ2 we immediately conclude from the finiteness of the r.h.s. of (10.38) that

j aµB(x) = j a

µR(x)

up to a finite multiplicative constant which is fixed as one by the current commuta-tion relations. Thus the conserved currents are not subject to renormalization, forexample,

j aµB = �BT aγµ�B = Z2�RT aγµ�R = j a

µR (10.41)

In the case of explicit symmetry breaking by soft (d < 4) terms one has∂µ jµ(x) �= 0 but, as discussed before, (10.40) can still be assumed. Writing ingeneral j a

µB = Z j jaµR and ∂µ j a

µB = ZD∂µ j a

µR we first conclude, arguing as before,that Z j = ZD. Next, Fourier transforming (10.38) and taking the limit p = 0 (pis conjugate to x) we get ZD = 1 since in the limit p = 0 the Fourier transformof the l.h.s. vanishes (explicit symmetry breaking accompanies the spontaneoussymmetry breaking and therefore there are no massless bosons in the spectrumcoupled to j a

µ currents). In conclusion, even if the symmetry is broken explicitlybut only softly the currents and their divergences are not subject to renormalization.For instance, for chiral symmetry broken by a mass term

L1 = −m�� (d = 3) (10.42)

for the divergence of the axial currents j a5µ we have (from the equations of motion)

∂µ j a5µ = 2mi�T aγ5� (10.43)

and the non-renormalization theorem implies

Zm = Z−1�γ5�

(10.44)

where

(�γ5T a�)B = Z�γ5�(�γ5T a�)R (10.45)

Absence of wave-function renormalization for conserved or partially conservedcurrents does not, of course, exclude subtractive renormalizations for time-orderedproducts of such currents, as discussed earlier.

10.2 Quark masses and chiral perturbation theory 353

10.2 Quark masses and chiral perturbation theory

Simple approach

Using the Ward identity for the Green’s function involving the axial current Aaµ(x)

and the pseudoscalar density �a(x) we have proved in Section 9.5 that in chirallysymmetric QCD with chiral symmetry spontaneously broken there exist masslessGoldstone particles in the physical spectrum. The same Ward identity gave us therelation (9.101) between the parameters fπ and 〈��〉 of spontaneous breaking ofchiral symmetry

Z fπ = −〈��〉 (9.101)

where the constants fπ and Z are defined by (9.97) and (9.98)

〈0|�a(x)|πb(p)〉 = Zδab exp(−ipx)

and

〈0|Aaµ(x)|πb(p)〉 = i fπδ

ab pµ exp(−ipx)

respectively. (In view of our discussion in the previous section check, using theOPE for TAa

µ(x)�b(y), that the Ward identity (9.91) does not need subtractions

and therefore in the chiral limit the relation (9.101) is free of any subtraction-dependent parameters.)

Imagine now a realistic case in which the chiral symmetry is also brokenexplicitly by the mass term (10.42). Then the proof of Goldstone’s theorem breaksdown because

∂µAaµ(x) = 2mi�T aγ5� �= 0 (10.43a)

Instead, one can now calculate masses of the would-be (pseudo-) Goldstone bosonsin terms of the explicit symmetry breaking parameter: quark mass m. Suchcalculation is based on perturbative treatment of the symmetry breaking interaction(chiral perturbation theory) but takes account of all orders of the interactionsdescribed by the chirally symmetric part L0 of the lagrangian.

It is worth mentioning at this point that the subject of chiral perturbation theory isnot limited to the problem of the pseudoscalar meson masses but covers the wholelow energy physics of pions and kaons. Several techniques have been developedto expand the bound state masses, Green’s functions and any other quantity in thelow energy pseudoscalar meson sector (which in the limit of exact chiral symmetrydepend on only two parameters fπ and 〈��〉) in powers of the light quark massesmu,md and ms (Dashen 1969, Dashen & Weinstein 1969, Pagels 1975). The mostsystematic approach proposed in a very interesting series of papers (Leutwyler1984, Gasser & Leutwyler 1984, 1985a, b) relies on the effective lagrangiantechnique (see Chapter 14).


Here we shall only be concerned with the pseudoscalar meson masses calculatedin the first order in the quark mass m. The answer is almost immediate.Eq. (10.43a) sandwiched between the vacuum and the one-pion state together withthe definitions (9.97) and (9.98) imply the exact relation

fπm2π = 2m Z (10.46)

Working to the first order in m we can now use (9.101), valid in the chiral limit, toget

m2π = −(1/ f 2

π )2m〈0|��|0〉 (10.47)

which is the desired result valid to the lowest order in chiral perturbation theory. Itis the pion mass, for example, in SU (2)× SU (2) symmetric QCD with � = (u, d)and mu = md = m. The pion mass has been expressed in terms of the parametersof chiral symmetry breaking: spontaneous 〈0|��|0〉 and explicit m.

Our discussion is always in terms of renormalized quantities. We recall thatinteractions described by the chirally invariant part L0 of the lagrangian areincluded to all orders in our discussion. Therefore m = m(µ), where µ is somerenormalization point, and operators �γ5T a� and �� are also renormalized at thesame point. However, as discussed in Section 10.1 the products m�γ5T a�,m��

are renormalization-scheme-independent.The result (10.47) can be extended to the case of three flavours with mu �= md �=

ms (for heavy flavours the breaking of chiral symmetry by the mass terms is toostrong to be reliably considered as a perturbation). We get the following:

m2π0 = m2

π± = (mu + md)C + O(m2q)

m2K+ = (mu + ms)C + · · ·

m2K0 = m2

K0 = (md + ms)C + · · ·C = −(2/ f 2

π )〈0|uu|0〉, 〈0|uu|0〉 = 〈0|dd|0〉 = 〈0|ss|0〉

(10.48)

To extract reliable values for the quark mass ratios we need, however, to includethe electromagnetic mass corrections which will be considered in Section 10.4.

Finally we would like to stress that the quark masses we talk about are therenormalized quark mass parameters of the lagrangian (the so-called currentquark masses). For a mass-independent renormalization procedure ratios of therenormalized masses do not depend on µ.

Approach based on use of the Ward identity

The result (10.47) can also be derived using a Ward identity directly for the realisticcase with the chiral symmetry broken explicitly. This approach can be used in other

10.2 Quark masses and chiral perturbation theory 355

problems too and its generalizations are used to calculate higher order correctionsin the chiral perturbation theory.

The starting point is the Ward identity for the Green’s function〈0|TAa

µ(x)∂νAb

ν(0)|0〉, where Aaµ(x) is the axial current and ∂νAb

ν standsfor 2mi�γ5T b�; similar considerations hold for any other current correspondingto a generator broken both spontaneously and explicitly when according to (10.23)

∂µ j aµ(x, t) = i[H′(x, t), Qa(t)] (10.23)

The Ward identity reads

∂µ

(x)〈0|TAaµ(x)∂

νAbν(0)|0〉 = Dab

5 (x)+ δ(x0)〈0|[Aa0(x), ∂

νAbν(0)]|0〉 (10.49)

where

Dab5 (x) = 〈0|T ∂µAa

µ(x)∂νAb

ν(0)|0〉The divergence ∂µAµ is soft: the identity (10.49) can then be regarded as anidentity for the renormalized Green’s functions.

Defining the Fourier transform ab5 (q) of the Dab

5 (x)

ab5 (q) = i

∫d4x exp(iqx)Dab

5 (x) (10.50)

Fourier transforming both sides of (10.49) and taking the limit q → 0 one gets thefollowing relation:

0 = ab5 (q = 0)+ i〈0|[Qa

5(t = 0), ∂νAbν(0)]|0〉 (10.51)

which in a general case, using (10.23), can also be written as

ab5 (q = 0) = −〈0|[Qa

5(t = 0), [Qb5(t = 0),H′(0)]]|0〉 (10.52)

For us H′ = m��. In the limit q → 0 the l.h.s. of (10.49) is zero because inthe present case (explicitly broken symmetry) there are no massless bosons in thephysical spectrum and therefore no poles at q2 = 0. The relation (10.52) can beused to calculate the would-be Goldstone boson (pion) masses in the first order inchiral perturbation theory if we approximate the Fourier transform ab

5 (q) by thepion pole. In general ab

5 (q) is given by the pole and the continuum contributionstarting at s = 9m2

π

ab5 (q2) = f 2

π m4πδ

ab

m2π − q2

+∫ ∞

(3mπ )2ds

ρ(s,m2π)

s − q2(10.53)

We have used 〈0|∂µAaµ(0)|πb〉 = fπm2

πδab and have written (10.50) as the spectral


integral using the standard technique (see, for example, Bjorken & Drell (1965)).For q2 = 0 one gets

ab5 (q2 = 0) = f 2

π m2πδ

ab + O(m4π) (10.54)

The leading term in m2π is given by the pion pole because this is the only term where

in the chiral symmetry limit the zero of the numerator in (10.53) partially cancelswith the denominator. The pion mass measures the departure from chiral symmetryinduced by perturbation H′(x), so expansion in m2

π is the chiral perturbation theory.In the order O(m2

π) we get the so-called Dashen’s relation

δabm2π = − 1

f 2π

〈0|[Qa5(t = 0), [Qb

5(t = 0),H′(0)]]|0〉 (10.55)

where fπ and |0〉 should be taken, for consistency, as in the chiral limit. In ourspecific problem, from commutation relations (9.90), we easily get (10.47).

It is also worth observing that the Green’s function Dab5 (x) or equivalently the in-

tegral in the spectral representation (10.53) does require subtractions. For instance,it can be seen from the OPE (see Chapter 7) that the product T ∂µAa

µ(x)∂νAb

ν(0)has a singularity O(m2x−6). The subtraction ambiguity is O(m2) or according toour discussion following (10.54) it is O(m4

π) and does not affect the first orderresults. However, in higher orders the subtraction constants appear as additionalfree parameters.

10.3 Dashen’s theorems

Formulation of Dashen’s theorems

We have learned in Section 9.4 that if a global symmetry G of a theory isspontaneously broken by the vacuum state |0〉 to a subgroup H then there is aset of degenerate vacua corresponding to different orientations of H in G. This setof vacua is generated by the action of the G transformations on the state |0〉

|�(g)〉 = U (g)|0〉 (10.56)

where U (g) is a unitary operator corresponding to a group element g. If thevacuum |0〉 is left invariant by H then |�(g)〉 is left invariant by the equivalentG-rotated subgroup gHg−1. The set of degenerate vacua is isomorphic to the cosetG/H space. If we introduce generators Xa and Y a which are, respectively, brokenand unbroken by the |0〉 then the transformation (10.56) can be written as

|�(�)〉 = exp(−i�a Xa)|0〉 (10.57)

Transformations in G/H space correspond to variations of the vacuum expectationvalues for which the effective action is level at its minimum. Zero mass particles

10.3 Dashen’s theorems 357

are quantized excitations – one for each orthogonal direction in G/H (see (9.51)and Chapter 14).

A small symmetry breaking perturbation H′(x) may change the above picturein a qualitative way: it may lift the degeneracy of the vacuum. The criteria foridentifying the correct vacuum in the presence of a small perturbation H′(x) arecalled Dashen’s theorems, and generally speaking, ensure the vacuum stability.To the leading order in the perturbation H′(x) the energy density in each of thedegenerate vacua is given by

E(�) = 〈�(�)|H′(0)|�(�)〉 = 〈0| exp(i�a Xa)H′(0) exp(−i�a Xa)|0〉(10.58)

To identify |0〉 as the true vacuum in the presence of the perturbation H′(x), E(�) should have a minimum (at least a local one) for � = 0. We thereforeget the following two conditions:

∂

∂�a E(�)|�=0 = i〈0|[Xa,H′(0)]|0〉 = 0 (10.59)

∂2

∂�a∂�b E(�)|�=0 = −〈0|[Xb, [Xa,H′(0)]]|0〉 ≥ 0 (10.60)

In view of the relation ∂µ j aµ(x) = i[H′(x), Qa], condition (10.59) also follows

from the requirement of the translational invariance of the vacuum: the vacuumvalue of a divergence then vanishes. Since charges Xa which do not annihilate thevacuum couple to Goldstone bosons 〈0|Xa|πa〉 �= 0, (10.59) says that 〈πa|H′|0〉 =0, i.e. the Goldstone boson tadpoles

vanish. Eq. (10.60) has a straightforward interpretation too: up to the renormaliza-tion factor the second derivative of the potential with respect to � is the Goldstoneboson mass matrix (see also Chapter 14) and (10.60) ensures that in the first orderperturbation in H′(x) the pseudo-Goldstone boson masses m2

ab are positive. So werederive Dashen’s formula (10.55) referring, however, to the previous derivationfor the fixing of the normalization factor.

Let us first see what the conditions (10.59) and (10.60) give for the σ -modelwith chiral symmetry explicitly broken by the H′(x) = +εσ ′ term. From thecommutation relations (9.30) we get

〈0|π |0〉 = 0 −ε〈0|σ ′|0〉 ≥ 0 (10.61)

for the correct vacuum, in agreement with (9.36). Using the normalization given in(10.55) we also rederive the result (9.39) for the pseudo-Goldstone boson mass.


For the SUL(2) × SUR(2) symmetric QCD lagrangian with chiral symmetrybroken spontaneously to SU (2) by a condensate UR〈�〉U+

L , where 〈�〉 = v1l as in(9.87) and explicitly by the mass term H′ = m�� one gets

〈0|�γ5�|0〉 = 0 −〈0|��|0〉 > 0 (10.62)

This follows from the commutation relation (9.88) and from the relations �� =Tr[�+�†] and �γ5� = Tr[�−�†], where � is defined by (9.84). In particularwe recover (9.86) which we have taken as the definitions of the vacuum in the chirallimit and in a specified basis for the chiral fields. Explicit symmetry breaking liftsthe degeneracy and leaves (10.61) and (10.62) as the only possibilities.

Dashen’s conditions find very interesting applications in models of the dy-namical breaking of the gauge symmetry of weak interactions (technicolourand extended technicolour models). Some of these ideas will be discussed inSection 11.3. Here we give a general technical introduction to such problemswhich will also be useful in the next section where we calculate the π+ − π0

mass difference.

Dashen’s conditions and global symmetry broken by weak gauge interactions

A physically very interesting case of an explicit global symmetry breaking by asmall perturbation is the following one: consider a strongly interacting theory (agauge theory) with some exact global symmetry G, for example, the chiral sym-metry which is spontaneously broken. Imagine then that some of the G-symmetrycurrents couple to some of the gauge bosons of the group GW of weak interactions(in the rest of this section all sums over the group indices are explicitly writtendown)

L1 = −∑α

gα Aαµ(x) jµα (x) (10.63)

where jµα (x) are linear combinations of some of the G currents; operators corre-sponding to quantum numbers commuting with G are also possible but irrelevantfor our discussion. In the first non-vanishing order of perturbation theory in gα

the G-symmetry breaking term of such a theory is given by the following effectivehamiltonian:

H′(0) = −∑α,β

12 gαgβ

∫d4x

µναβ (x)T jαµ(x) jβν (0) (10.64)

where µναβ (x) is the GW gauge boson propagator

µναβ (x) = i〈0|T Aµ

α (x)Aνβ(0)|0〉 ≡ δαβ µν(x) (10.65)


The question is whether our considerations in Section 10.1 based on the assumedsoftness of H′ apply to this case when the symmetry breaking term H′(0) is notsoft (d = 6), and at first glance it appears that it could affect the wave-functionrenormalization of fields. It turns out, however, that this does not necessarilyhappen. The problem can be studied by means of the operator expansion of theproduct of the two currents in (10.64). First of all the counterterms affectedby H′(0) are easily enumerated (by assumption H′(0) is used only in the firstorder perturbation): they involve only those operators of the expansion whichare accompanied by coefficients that produce non-integrable singularities. It isclear from (10.64) and (10.65) (remember that µν(x) ∼ 1/x2 for x → 0) thatthese are operators of dimension ≤ 4. The higher dimension operators in theexpansion of jαµ jβν do not cause a non-integrable singularity at x = 0, do notrequire renormalization and are irrelevant for our discussion; notice the differencein the case if H′(0) was considered in any order of perturbation theory. Among theoperators of dimension 4 there may be singlets under G which are again irrelevant.The non-singlets with d = 4 would affect our discussion but in several interestingcases they do not appear (see, for example, Section 10.4 for the breaking of thechiral group by H′ given by the product of two electromagnetic currents). Weassume here that the expression (10.64) converges without subtractions which arenon-singlets under the group G.

With perturbation H′(0) given by (10.64), Dashen’s conditions (10.59) and(10.60) can be rewritten in a more useful form if we express the vacuum energydensity E(�) given by (10.58) in terms of the G currents (Peskin 1981, Preskill1981). We recall that (10.58) and (10.64) together give the following expression

E(�) = − 12

∫d4x µν(x)

∑α

g2α〈0|T U † jαµ(x)UU † jαν (0)U |0〉 (10.66)

where U is any unitary operator representing the group G. The GW currents arelinear combinations of the G currents

gα jµα (x) =∑

a

gaα jµa (x) (10.67)

(α is the GW index, a is the G index and there is no sum over α on the l.h.s.) andconsequently

gαU † jµα (x)U =∑

a

gaαU † jµa (x)U =

∑a,c

gaα Rac(�) jµc (x) (10.68)

where R(�) is the adjoint representation of G. Because the vacuum is H -invariant(G → H spontaneously) only the H -invariant part of the product j c

µ j dν contributes

to (10.66) expressed in terms of (10.68). For any given vacuum from the degenerateset the generators of G split into two groups: the generators Y a such that Y a|0〉 = 0


and Xa where Xa|0〉 �= 0. Correspondingly, each G current carries index Y or X .Since Y a span a single real representation of H (the adjoint representation), bySchur’s lemma the only H -invariant term in the product Yj c

µYjd

ν is proportional toδcd :

〈0|T Yjcµ(x)

Yjdν (0)|0〉 = δcd〈0|T Yjµ

Yjν |0〉 = Tr[Y cY d]〈0|T YjµYjν |0〉 (10.69)

(here we normalize the broken and unbroken generators to Tr[Y aY b] =Tr[Xa Xb] = δab,Tr[Y a Xb] = 0), where Yjµ denotes any single current Yja

µ; thereis no summation in T YjµYjν . We will limit our discussion to cases in which in theproduct Xa Xb also the only H -invariant term is δab. It can be shown (Peskin 1981)that this restriction implies the existence of a parity operation P which preservesthe Lie algebra of G, such that

PY a P−1 = Y a, P Xa P−1 = −Xa (10.70)

For chiral symmetry (10.70) are, of course, true since P QaR,L P−1 = Qa

L,R.Eqs. (10.70) also imply that the joint expectation values of one Yja

µ and one Xjbν

are zero by parity conservation. The decomposition in (10.68) can be split into theY part and the X part

U †gα jµα (x)U = U †gα jµα (x)U |Y +U †gα jµα (x)U |x≡ YjµαU (x)+ XjµαU (x) (10.71)

(the last line defines our notation) such that only Y (X) currents from the sum∑a,c ga

α Rac j cµ contribute to the first (second) component. Taking all this into

account we obtain for E(�) the following:

E(�) = − 12

∫d4x µν(x)

∑α

{Tr[YJαUY JαU ]〈0|T Yjµ(x)

Yjν(0)|0〉

+Tr[XJαUX JαU ]〈0|T X jµ(x)

Xjν(0)|0〉} (10.72)

where JαU is the linear combination of the G generators corresponding to thecurrent U †gα jαµ(x)U with gα included. From the orthogonality of Xa and Y b

(Tr[XaY b] = 0)

Tr[YJαUY JαU ] = Tr[YJαU JαU ]

Tr[XJαUXJαU ] = Tr[XJαU JαU ]

}(10.73)

Using in addition an obvious relation

Tr[YJαU JαU ] = Tr[ JαU JαU ] − Tr[XJαU JαU ] (10.74)


we finally get

E(�) = − 12

∫d4x µν(x)

∑α

{Tr[ JαU JαU ]〈0|T Yjµ(x)Yjν(0)|0〉}

− 12

{∫d4x µν(x)〈0|T [Xjµ(x)

Xjν(0)− Yjµ(x)Yjν(0)]|0〉

}×∑α

Tr[XJαUXJαU ] (10.75)

The first term in the sum is �-independent (without subtraction it may even beinfinite) and in the second term �-dependence has been factored out from thecurrent matrix elements. Assuming that the coefficient of the last trace in (10.75)is negative as can be argued in some cases, one gets the preferred vacuum simplyby minimizing the quantity ∑

α

Tr[XJαUXJαU ]

i.e. by minimizing the trace for projections of currents Jα on the subspace ofpossible equivalent representations of the Xa and Y a or, regarding the vacuum|0〉 as fixed so that the rotation is applied to the perturbation H′, for projectionsof transformed currents on the subspace of the original broken generators. Onesees that the preferred vacuum orientation corresponds to the minimal number ofbroken generators in the decomposition of J α into the generators of G.

We can rewrite Dashen’s conditions (10.59) and (10.60) in the present notation.Writing U = exp(−i�a Xa) one has

∂

∂�a

∑α

Tr(XJαU )2|U=1 = 2i∑α

Tr[X [Xa, Jα]XJα] = 2i∑α

Tr[Xa[YJα,X Jα]]

(10.76)where the last result follows from the commutation relations [X, X ] = iY follow-ing from (10.70), orthogonality of Y and X and the trace property Tr[[A, B]C] =Tr[A[B,C]]. Thus, the vacuum orientation is a stationary point of E if[YJα,X Jα] = 0 for all αs.

The second Dashen’s condition reads

m2ab =

1

2

∂2

∂�a∂�b

∑α

Tr(XJαU )2|U=1 × M2 ≥ 0 (10.77)

where, from (10.75)

M2 = (1/ f 2π )

∫d4x µν(x)〈0|T [Yjµ(x)

Yjν(0)− Xjµ(x)Xjν(0)]|0〉 (10.78)

and Yjµ (Xjµ) denotes, as before, any single current corresponding to an unbroken


(broken) generator of G. The �-dependence is only in the trace which can beexpressed as follows

∂2

∂�a∂�b

∑α

T r(XJαU )2|U=1

= −2∑α

{Tr[X [Xa, [Xb, Jα]]XJα] + Tr[X [Xa, Jα]X [Xb, Jα]]}

= 2∑α

Tr[[YJα, [YJα, Xa]]Xb − [XJα, [XJα, Xα]]Xb] (10.79)

Dashen’s condition (10.60) or equivalently the positivity of the pseudo-Goldstonebosons mass matrix depends on the signs of the trace (10.79) and of the integral(10.78). The latter can often be studied by means of the spectral function sumrules. Several applications of the results of this section will be discussed in thenext section and in Section 11.3.

10.4 Electromagnetic π+–π0 mass difference and spectral function sum rules

Electromagnetic π+–π0 mass difference from Dashen’s formula

In Section 10.2 we have studied an explicit chiral symmetry breaking in QCDcaused by non-zero current quark masses. Another source of such a breaking isthe electromagnetic interaction of quarks which couples the T 3 generator of theisospin group to the electromagnetic gauge field: Q = T 3 + Y . The effectivehamiltonian is given by (10.64) with jµ(x) being the electromagnetic current and µν the photon propagator. Following our discussion in Section 10.3 we noticethat the expansion of the operator product T jµ(x) jµ(x) reads

T jµ(x) jµ(x) = C1(x)1l +∑

q

Cq(x)mqqq + CG(x)GαµνGµν

α

+ operators with d > 4 + non-scalar operators (10.80)

where Gαµν is the gluon field strength. Only scalar operators contribute to the

integral in (10.64). Thus, the only d = 4 operator (GαµνGµν

α ) is a singlet underthe chiral group and our considerations of Sections 10.1 and 10.3 apply. We alsosee that in the chiral limit mq = 0 all the divergent terms in Hem are singlets underthe chiral group and may be disregarded: they induce a universal energy shift forall states, including the vacuum.

This section is devoted to a study of the electromagnetic contribution to thepion masses. The quark masses will be neglected in the calculation of theelectromagnetic masses in the spirit of the first order perturbation in any explicitsymmetry breaking term. As mentioned, in this case the electromagnetic pion mass

10.4 π+–π0 mass difference 363

difference is finite without renormalization. The calculation can be performed bymeans of Dashen’s formula.

Using (10.77), (10.79) and the fact that J αY = eT 3, J α

X = 0 for the electromag-netic current (as in Section 9.6 we define the vacuum of the SUL(2) × SUR(2)symmetric QCD so that the SUL+R(2) remains unbroken; then J α

X = 0 andDashen’s condition for the stationary vacuum is satisfied in the presence of theelectromagnetic interactions) we immediately get the following result for theone-photon exchange contribution to the pion mass:

(m2π+)

γ = e2 M2, (m2π0)

γ = 0 (10.81)

where M2 can be explicitly rewritten in terms of the vector and axial currents

M2 = (1/ f 2π )

∫d4x µν(x)〈0|T [V3

µ(x)V3ν (0)−A3

µ(x)A3ν(0)]|0〉 (10.82)

We recall again that all, possibly infinite, singlet contributions to the electromag-netic masses have been disregarded in (10.81) and (10.82) (see the derivation of(10.77) in the previous section). We also remember that both equations are validonly in the first order in the symmetry breaking, i.e. for mq = 0. Non-zero quarkmasses introduce second order corrections O(αmq) which need renormalizationand are therefore renormalization-scale-dependent. Formula (10.82) was derivedby Das et al. (1967). To calculate the integral (10.82) one can use the spectralfunction sum rules.

Spectral function sum rules

Spectral function sum rules, originally derived by Weinberg (1967a) for theSU (2) × SU (2) chiral symmetry, can be discussed in a general way by invokingthe OPE (Bernard, Duncan, Lo Secco & Weinberg 1975). Let us consider theKallen–Lehmann spectral representation (Bjorken & Drell 1965) for the time-ordered product of two currents

〈0|T jaµ(x) j b

ν (0)|0〉 =∫ ∞

0dµ2

∫d4k

(2π)4exp(−ikx)

i

k2 − µ2 + iερabµν(k, k2 = µ2)

(10.83)


where the spectral function ρ can be decomposed into the spin-one and spin-zeroparts:

(2π)3∑

nspin 1

〈0| j aµ(0)|n〉〈n| j b

ν (0)|0〉δ(k − kn) = −(gµν − kµkν/µ2)ρ(1)ab(µ2)

(10.84)

(2π)3∑

nspin 0

〈0| j aµ(0)|n〉〈n| j b

ν (0)|0〉δ(k − kn) = kµkνρ(0)ab(µ2) (10.85)

Taking the Fourier transform of (10.83)

F.T. =∫

d4x exp(ikx)〈0|T jaµ(x) j b

ν (0)|0〉 = i∫ ∞

0dµ2

ρabµν(k, k2 = µ2)

k2 − µ2

(10.86)

and expanding it formally in powers of 1/k2 we get

F.T. = ikµkν

k2

∫ ∞

0dµ2

[ρ(1)ab (µ2)

µ2+ ρ

(0)ab (µ2)

]− i

gµν

k2

∫ ∞

0dµ2 ρ

(1)ab (µ2)

+ ikµkν

(k2)2

∫ ∞

0dµ2 [ρ(1)

ab (µ2)+ µ2ρ(0)ab (µ2)] + · · · (10.87)

This formal expansion can be meaningful only when the Fourier transform∫d4x exp(ikx)〈0|T ja

µ(x) j bν (0)|0〉 behaves for large k2 as O(1) or softer. Otherwise

the coefficients of the expansion must be divergent. If one constructs a linearcombination of currents such that the Fourier transform (10.86) is for large k2

softer than O(1), then the first few terms in the expansion (10.87) must vanish,for example, for softer than (1/k2) behaviour∫ ∞

0ds [ρ(1)

ab (s)/s + ρ(0)ab (s)] = 0∫ ∞

0ds ρ

(1)ab (s) = 0∫ ∞

0ds sρ(0)

ab (s) = 0

(10.88)

and these are the spectral function sum rules.The OPE is useful in finding appropriate combinations of current products with

soft high energy behaviour. In looking for such products one can use the globalsymmetry of the lagrangian which is respected by the OPE coefficient functionseven if the symmetry is spontaneously broken. In our problem we are interested in


the group G = SUL(2)× SUR(2) broken to the H = SUL+R(2). Spectral functionsum rules can be derived by studying the high momentum limit of

Gabµν(k) =

∫d4x exp(ikx)〈0|T ja

µL(x) j bνR(0)|0〉

which transforms as the adjoint representation (3, 3) under SU (2) × SU (2).The asymptotic behaviour of Gab

µν(k) is determined by the lowest-dimensionoperator in the expansion of j a

Lµ j bRν which has a non-zero vacuum expectation

value. This operator must be Lorentz-invariant, gauge-invariant, H -invariant,i.e. a singlet under SUL+R(2), and, because the expansion coefficient functionsrespect the G symmetry, it must transform as the adjoint representation (3, 3) underSU (2) × SU (2). The lowest-dimension operator satisfying these requirements isa four-fermion operator of dimension (mass)6 (�� transforms as (2, 2)⊕ (2, 2)).Therefore, Gab

µν ∼ (k2)−2 up to logarithms and the sum rules (10.88) hold for thespectral functions ρab

LR.According to (10.82) we are interested in the combination VV −AA = 4 jL jR.

Taking into account that

〈0|Aaµ|πb〉 = ikµ fπδ

ab

and therefore

ρ(0)Aab(µ

2) = δab f 2π δ(µ

2)+ · · ·and neglecting any continuum contribution to ρ

(0)A which should be small we get

from (10.88) the following sum rules:∫ ∞

0(ds/s)[ρ(1)

Vab(s)− ρ(1)Aab(s)] = f 2

π δab∫ ∞

0ds [ρ(1)

Vab(s)− ρ(1)Aab(s)] = 0

(10.89)

which are valid in the chirally symmetric theory. If the chiral symmetry is brokenby non-zero quark masses the OPE for the difference VV − AA contains a termproportional to mqq . This can be seen by a spurion analysis (Wilson 1969a): thecurrent product belongs to (3, 3) of the SU (2) × SU (2), whereas the operatorqq belongs to (2, 2) ⊕ (2, 2). However, the symmetry is broken by the massterm mqq . Representing the symmetry breaking interaction by a spurion whichmust also belong to (2, 2)⊕ (2, 2) we see that combining one spurion with the qqoperator one can produce the (3, 3) representation. Thus, in the expansion of thetwo-current product we need one power of the symmetry breaking parameter m forthe coefficient function in front of the operator qq to be non-zero. The second sumrule is then no longer valid: the coefficient of this mqq term scales as x−2, so its


Fourier transform behaves as k−2. The corrections are, however, expected to besmall because the masses mu and md are very small in comparison to the scale ofQCD.

Results

We are now ready to calculate M2, (10.82). Introducing explicitly

µν(x) = gµν

∫d4k ′

(2π)4

exp(−ik ′x)k ′2 + iε

we get

M2 = − i

f 2π

∫ ∞

0dµ2 3[ρ(1)

V (µ2)− ρ(1)A (µ2)]

∫d4k

(2π)4

1

k2(k2 − µ2)(10.90)

The momentum integral is logarithmically divergent and gives, using dimensionalregularization,

1

(4π)2

1

ε+ 1

(4π)2ln

µ20

µ2+ const. (10.91)

However, it follows from the second sum rule (10.89) that the coefficient of thedivergent term vanishes. This just reflects the already discussed fact that the firstorder electromagnetic contribution is finite after the singlet piece under SU (2) ×SU (2) has been disregarded. The remaining finite term gives

(m2π+)

γ = 3α

4π( fπ)2

∫ ∞

0ds

(ln

µ20

s

)[ρ(1)

V (s)− ρ(1)A (s)] (10.92)

and on account of the second sum rule (10.89) it is renormalization point (µ20)

independent. Finally we can calculate (10.92) assuming that it is reasonable tosaturate the integral (10.92) with the two lowest-lying narrow resonances, one (ρ)

coupled to the vector current and the other (A1) to the axial current. Taking

ρV(s) = f 2ρ sδ(s − m2

ρ)

ρA(s) = f 2A1

sδ(s − m2A1)

and using both sum rules (10.89) one gets

f 2ρ = f 2

A1+ f 2

π


and

m2ρ f 2

ρ = f 2A1

m2A1

Therefore we can eliminate the A1 parameters to get the final answer in terms ofthe ρ parameters:

(m2π+)

γ = 3α

4πm2

ρ

(fρfπ

)2

lnf 2ρ

f 2ρ − f 2

π

(10.93)

To calculate the π+–π0 mass difference we have to combine the results ofSection 10.2 and of this section. In the first order in both symmetry breakingperturbations, quark masses and electromagnetic interactions, we have, using thefirst equation (10.48),

m2π+ = m2

π0 + (m2π+)

γ (10.94)

where

m2π0 = C(mu + md), C = −(2/ f 2

π )〈0|uu|0〉

and therefore

mπ+ − mπ0 ≈ (m2π+)

γ /2mπ0

Using fπ = 93 MeV and fρ = (145 ± 8) MeV, one gets mπ+ − mπ0 = (4.9 ±0.2) MeV. Although the error due to the resonance saturation is hard to estimateaccurately this result should be regarded as at least in qualitative agreement withthe experimental value 4.6 MeV.

Corrections to this result due to the π0–η mixing caused by mu �= md can beestimated to be negligibly small (Gasser & Leutwyler 1982). Finally, as alreadymentioned, corrections of order αmq require renormalization but are also expectedto be small (Gasser & Leutwyler 1982).

Extending our considerations to the SU (3) × SU (3) case we get (m2K+)

γ =(m2

π+)γ and (m2

K0)γ = 0. Therefore, again using (10.48),

m2K+ = C(mu + ms)+ (m2

π+)γ

m2K0 = C(md + ms)

}(10.95)

and the K+–K0 mass difference calculated in the first order of the explicit chiralsymmetry breaking is not purely electromagnetic but it also reflects a possible mu–md difference. Eqs. (10.94) and (10.95) taken together can be used to estimate thequark mass ratios (Weinberg 1977). Completing our discussion given at the end of


Section 10.2 we get

md

mu= m2

K0 − m2K+ + m2

π+

2m2π0 + m2

K+ − m2K0 − m2

π+= 1.8

ms

md= m2

K0 + m2K+ − m2

π+

m2K0 − m2

K+ + m2π+

= 20.1

We stress that these results are valid in the first order of the chiral symmetrybreaking.

11

Higgs mechanism in gauge theories

11.1 Higgs mechanism

The proof of Goldstone’s theorem given in Chapter 9 for theories with sponta-neously broken global symmetry requires Lorentz invariance and Hilbert spacewith positive-definite scalar products. Gauge theories do not obey both require-ments simultaneously. In a covariant gauge, the theory contains states of negativenorm. In a gauge in which the theory has only states of positive norm it is notmanifestly covariant. In consequence, Goldstone’s theorem does not hold and theso-called Higgs mechanism operates (see, for instance, Abers & Lee (1973) andreferences therein).

We shall consider the simplest example of the U (1) gauge theory. This exampleillustrates all the essential points. The Higgs mechanism requires the presence of ascalar field coupled to the gauge field. For a U (1) symmetric theory it has to be acomplex scalar field.

The lagrangian for a single self-interacting complex field �′

L = ∂µ�′∗∂µ�′ − λ(�′∗�′ + m2/2λ)2 (11.1)

is invariant under global U (1) transformation: U�′ = exp(−i�)�′. The U (1)symmetry is spontaneously broken (see Section 1.2) for m2 < 0. For the real fields�′ = (ϕ′ + iχ ′)/

√2 we can choose, for instance,

〈0|ϕ′|0〉 = v = (−m2/λ)1/2, 〈0|χ ′|0〉 = 0

and defining the physical fields ϕ′ = ϕ − v, χ = χ ′ we get

L = 12(∂µϕ∂

µϕ + ∂µχ∂µχ)− v2λϕ2 − 14λ(ϕ

2 + χ2)2 − λϕv(ϕ2 + χ2) (11.2)

This lagrangian describes the interaction of the massive particle ϕ and the masslessGoldstone boson χ .

369

370 11 Higgs mechanism in gauge theories

It is also useful to introduce ‘angular’ variables and parametrize the field �′ asfollows (see (9.78))

�′(x) = ρ ′(x) exp[−iη′(x)T ]

(01

)=(

ρ ′ sin η′

ρ ′ cos η′

); T =

(0 i−i 0

)(11.3)

Using the freedom in our choice of the vacuum expectation value among the U (1)degenerate set (Section 1.2) we can write

�′(x)− 〈0|�′(x)|0〉 = [ρ ′(x)− v] exp[−iη′(x)T ]

(01

)(11.4)

and in terms of the fields ρ = ρ ′ − v and η = vη′ the lagrangian (11.1) reads

L = 12(∂µρ∂

µρ + ∂µη∂µη)− v2λρ2 + cubic and higher order terms (11.5)

Thus, the particle interpretation of the fields ρ and η is the same as that of ϕ and χ .Now let us introduce a gauge field Aµ; a lagrangian invariant under local gauge

transformation is

L = − 14 Fµν Fµν + (Dµ�

′)∗(Dµ�′)− λ(�′∗�′ + m2/2λ)2 (11.6)

where

Dµ = ∂µ + ig Aµ

The gauge freedom has profound implications for the mechanism of spontaneoussymmetry breaking. This is easiest to see working with the angular variables.Choosing the gauge function to be −η′(x) = −η(x)/v and using (11.4) we get(in the real basis)

U�′(x)− 〈0|�′(x)|0〉U = [ρ ′(x)− v]

(01

)AU

µ(x) = Aµ(x)+ (1/gv)∂µη(x)

(11.7)

Since the lagrangian (11.6) is gauge-invariant we can write it in terms of fields U�

and AUµ and then

L = − 14 FUµν FU

µν + 12∂µρ∂

µρ + 12 g2(ρ + v)2 AU

µ AUµ − 14λ(ρ

2 + 2ρv)2 (11.8)

where

ρ = ρ ′ − v

In this gauge, called the unitary gauge (introduced in Section 1.3), the particlespectrum is evident. There is a scalar meson ρ with the bare mass (2λv2)1/2

and a massive vector meson with mass gv and no particle corresponding to the

11.1 Higgs mechanism 371

field η which has disappeared from the lagrangian. The η degree of freedomhas been traded for the longitudinal component of the vector field, now massive.Thus, the gauge symmetry is spontaneously broken and no massless Goldstoneboson remains in the physical spectrum. The Higgs mechanism is one of thefundamental ideas for condensed matter physics and it is also the crucial part ofunified electroweak theory (see Chapter 12).

Quantization in the unitary gauge gives a theory which is manifestly unitaryorder-by-order in perturbation theory but not manifestly renormalizable. Becauseof the massive vector meson propagator which for large k grows like kµkν/m2k2

the Green’s functions contain divergences that cannot be removed by the renormal-ization counterterms. Such divergences vanish in the S-matrix elements but this weknow from the proof of renormalizability of a spontaneously broken gauge theorygiven in covariant gauges (Abers & Lee (1973) and references therein). Hence it isuseful to extend our discussion to covariant gauges.

Let us first take a gauge fixed by the standard gauge-fixing term −(∂µ Aµ)2/2a.Returning to the variables ϕ and χ we get from (11.6) the following lagrangian:

L = − 14 Fµν Fµν + 1

2(∂µϕ∂µϕ + ∂µχ∂µχ)− λv2ϕ2 − (1/2a)(∂µ Aµ)2 (S0)

+ 12 g2v2 Aµ Aµ + gvAµ∂

µχ (S0)

− g Aµ∂µϕχ + g Aµ∂

µχϕ + 12 g2 Aµ Aµ(ϕ2 + 2ϕv)

+ 12 g2 Aµ Aµχ2 − 1

4λ(ϕ2 + χ2)2 − λ(ϕ2 + χ2)ϕv

}(SI)

(11.9)

One can now proceed in several different ways. One possibility is to take S0 asthe free lagrangian of the theory with the corresponding definitions of the freeparticle states (which may not be physical states). With the standard methods ofChapters 2 and 3 we get then the following free particle propagators (for example,in the Landau gauge a → 0):

Aµ(massless) : −i(gµν − pµ pν/p2)/(p2 + iε) ≡ −iDµν

ϕ(massive) : i/(p2 − 2λv2 + iε)χ(massless) : i/(p2 + iε)

(11.10)

The full gauge boson propagator

µν(p) = −iDµν

1

1 −�(p2)(11.11)

where �(p2) is defined by the gauge-invariant vacuum polarization tensor,

≡ i�µν(p2) = i(gµν p2 − pµ pν)�(p2) (11.12)

gets a contribution from the remaining terms in the lagrangian and in particular


from S0 which, although quadratic in fields too, is counted as an interaction term.In the lowest order the S0 contribution is

≡ −i(pµ pν/p2)g2v2 (11.13a)

≡ +ig2v2gµν (11.13b)

Hence

�(p2) = g2v2/p2 (11.14)

and consequently

µν(p) = −i(gµν − pµ pν/p2)/(p2 − g2v2) (11.15)

which can be rewritten as a combination of the standard propagator for a massivevector boson and of a term corresponding to a scalar negative norm boson coupledto the source of the vector meson gradiently

µν(p) = −i

(gµν − pµ pν

g2v2

)1

p2 − g2v2 + iε− i

pµ pν

g2v2

1

p2 + iε(11.16)

Note also that the χ propagator remains massless in the Landau gauge.One sometimes says that the coupling (11.13a) of a gauge boson to a massless

Goldstone boson is the origin of the gauge boson mass and of the spontaneousbreaking of gauge invariance. In this context one should observe that the masslessGoldstone boson contributes only to the pµ pν term in the �µν but by gaugeinvariance the gµν term must then be present. This remark will be particularlyrelevant in Section 11.3 where other mechanisms for spontaneous gauge symmetrybreaking are discussed. Secondly, such an interpretation is certainly procedure-and gauge-dependent. Indeed, we can as well take S0 + S0 as our free particlelagrangian. The presence of the mixing term gvAµ∂

µχ leads to a system ofcoupled differential equations for Aµ and χ propagators and in the limit a → 0 weget (11.15) as the free boson propagator. The coupling disappearsfrom the theory as well as its previous interpretation.

Finally we can get rid of the coupling gvAµ∂µχ in the lagrangian by a clever

choice of gauge. This so-called Rξ gauge is in wide use. Take the gauge-fixingterm as

f (Aµ, χ) = − 1

2ξ(∂µ Aµ − ξgvχ)2 (11.17)

Then, integrating by parts when necessary,

S0 + f (Aµ, χ) = 12 g2v2 Aµ Aµ − 1

2ξg2v2χ2 − (1/2ξ)(∂µ Aµ)2 (11.18)

11.2 Spontaneous gauge symmetry breaking 373

and the free propagators are

Aµ:−i

p2 − M2 + iε

[gµν − (1 − ξ)

pµ pν

p2 − ξ M2

]= −i

gµν − pµ pν/M2

p2 − M2 + iε− i

pµ pν/M2

p2 − ξ M2 + iε

χ : i/(p2 − ξ M2 + iε)

ϕ: i/(p2 − 2λv2 + iε)

where M = gv. In the limit ξ = 0 we get the Landau gauge and for ξ = 1 theso-called ’t Hooft–Feynman gauge. The unitary gauge is recovered in the limitξ →∞.

For any finite value of ξ there are unphysical poles at p2 = ξ M2 in the gaugeboson propagator and in the χ propagator. They cancel, however, in the S-matrixelements which are obtained from the Green’s functions by removing externallines, setting external momenta on the mass-shell and contracting tensor indiceswith appropriate physical polarization vectors. This can be shown, for instance, byproving that the renormalized S-matrix is independent of ξ (Abers & Lee 1973).

11.2 Spontaneous gauge symmetry breaking by radiative corrections

In the model of the previous section gauge symmetry has already been sponta-neously broken at the tree level. Here we discuss spontaneous gauge symmetrybreakdown by radiative corrections (Coleman & Weinberg 1973). This can occurwhen elementary scalar fields of zero mass are present in the theory (Georgi & Pais1977). In the tree approximation there is then no spontaneous symmetry breakingbecause in this approximation a non-zero vacuum expectation value of the scalarfields can appear only as a result of interplay between the �4 interaction termsand the scalar mass terms. Spontaneous symmetry breaking is, however, producedby non-zero vacuum expectation values induced by the higher order corrections.This is also an example of a dimensional transmutation: a dimensionful parameter〈0|�|0〉 is generated in a quantum field theory whose classical limit is describedby a scale-invariant lagrangian.

We consider first λ�4 theory with a massless meson field and discuss sponta-neous breaking of global reflection symmetry � → −�. The lagrangian densityfor this theory written in terms of the renormalized quantities

L = 12(∂µ�)(∂µ�)− (λ/4!)�4 + 1

2 A(∂µ�)(∂µ�)− 12 B�2 − 1/4!C�4 (11.19)

contains the usual wave-function and coupling-constant renormalization counter-terms. Also a mass renormalization counterterm is present, even though we are


studying the massless theory, because the theory possesses no symmetry whichwould guarantee vanishing bare mass in the limit of vanishing renormalized mass(the scale invariance is broken by anomalies). To study spontaneous symmetrybreaking in such a theory we will use the effective potential formalism developedin Section 2.6. To the one-loop approximation the effective potential is given by(2.165) and (2.166) (with m2 = 0 and with the contributions from the mass andcoupling-constant counterterms included)

V = λ

4!ϕ4 + 1

2 Bϕ2 + 1

4!Cϕ4 + 1

2

∫d4k

(2π)4ln

(1 + λϕ2

2k2

)(11.20)

In this expression we have performed the usual Wick rotation so the integral is inEuclidean space: d4k = 1

2 k2 dk2 d�4, �4 = 2π2. Observe also that, even thoughfor m2 = 0 the series in (2.166) is disastrously IR divergent, its sum exhibits onlya logarithmic singularity at ϕ = 0. Cutting off the integral in (11.20) at k2 = �2

(exceptionally we use the UV cut-off rather than the dimensional regularizationmethod), performing the elementary integration and throwing away terms whichvanish as �2 goes to infinity we obtain

V = λ

4!ϕ4 + 1

2 Bϕ2 + 1

4!Cϕ4 + λ�2

64π2ϕ2 + λ2ϕ4

256π2

(ln

λϕ2

2�2− 1

2

)(11.21)

Imposing now the definitions of the renormalized mass and coupling constant wedetermine the value of the counterterms. Defining the renormalized squared massof the field �(x) as the value of the inverse propagator at zero momentum andrecalling the interpretation following (2.164) of the derivatives of the effectivepotential we get

d2V/dϕ2|ϕ=0 = 0 (11.22)

as the mass renormalization condition. We take the derivative at ϕ = 0 becausewe are interested in the inverse propagator of the original (not shifted) field �(x);note also that the perturbative calculation of V (ϕ) makes use of the Feynman rulesfor the lagrangian written in terms of the field �(x) and therefore it is based on(2.164) even though the theory may be spontaneously broken. From (11.21) and(11.22) we get

B = −λ�2/32π2

A possible definition of the renormalized coupling constant would be to identifyit with the value of the 1PI four-point function in the original fields �(x) at zeroexternal momenta

d4V/dϕ4|ϕ=0 = λ

Unfortunately, for a massless field we encounter the logarithmic IR singularities.


In the standard renormalization procedure one defines in such a case the couplingconstant at some off-mass-shell position in momentum space. Using the effectivepotential formalism it is much more convenient to adopt an alternative definition,namely

d4V/dϕ4|ϕ=M = λ(M) (11.23)

where M is some arbitrary number with the dimension of mass. It is clear from ourdiscussion in Section 2.6 (see (2.156)) that the fourth derivative of V (ϕ) at ϕ = Mis the 1PI Green’s function, at zero external momenta, for the shifted fields �′(x) =�(x)− M . In particular if we take M = 〈0|�|0〉 it is the four-point function takenat zero external momenta for the physical massive field �(x): 〈0|�(x)|0〉 = 0.

Imposing condition (11.23) on the effective potential V (ϕ) given by (11.21) wefinally get

V = λ(M)

4!ϕ4 + λ2(M)ϕ4

256π2

(ln

ϕ2

M2− 25

6

)(11.24)

We see that the one-loop radiative corrections have turned the minimum at theorigin ϕ = 0 into a maximum and caused a new minimum of V to appear at

λ(M) ln(〈0|�|0〉/M) = − 323 π2 + O(λ) (11.25)

Thus, the one-loop corrections have generated spontaneous breaking of reflectionsymmetry. However, in our example, the new minimum lies outside the rangeof validity of perturbation theory as higher orders will give higher powers ofλ ln(ϕ2/M2) which according to (11.25) is large for ϕ = 〈0|�|0〉.

Physically interesting theories in which spontaneous symmetry breaking can bestudied in perturbation theory are gauge theories with elementary massless scalarfields and possibly fermion fields coupled to the gauge fields. Let us consider thecase of a single gauge field coupled to a charged scalar to see qualitatively whathappens. In the one-loop approximation the effective potential of the scalar sectorgets a contribution from the scalar loops which, aside from irrelevant numericalfactors (there are now two real scalar fields in the theory: � = (�1 + i�2)/

√2;

the scalar self-coupling is λ(��∗)2; the effective potential can only depend onϕ2 = (ϕ2

1 + ϕ22) because the theory is U (1)-invariant), have the same structure

as before, and from the gauge field loops in Fig. 11.1 arising from the minimalcoupling |(∂µ + ig Aµ)�|2. The trilinear terms lead to diagrams such as

which vanish if we work in the Landau gauge, where the gauge boson propagator


Fig. 11.1.

is

Dµν = −igµν − kµkν/k2

k2 + iε

This is because we calculate diagrams with zero external momenta, and thereforethe momentum of the internal scalar meson is the same as that of the internal gaugeboson.

It is obvious now that the gauge boson loop contribution to the effective potentialhas the same structure as the scalar loop contribution, so we get

V = λ(M)ϕ4 + [(1/16π2)λ2(M)+ bg4(M)]ϕ4[ln(ϕ2/M2)− 256 ] (11.26)

where b is some numerical factor. If it happens that λ(M) � g2(M) � 1 we canneglect the O(λ2) term and then dV/dϕ = 0 for

|〈ϕ〉| = (〈0|�1|0〉2 + 〈0|�2|0〉2)1/2 = M exp

[11

6− 1

2b

λ(M)

g4(M)

](11.27)

Thus, in the one-loop approximation we find in our theory spontaneous breakingof the U (1) symmetry and, this time, the new minimum is in the range of validityof our approximation: due to our choice of the renormalization point such thatλ(M) � g2(M), (11.27) constrains only the value of ln(〈ϕ〉2/M2) and for smallλ and g the terms λ ln(〈ϕ〉2/M2) and g2 ln(〈ϕ〉2/M2) are small. A non-vanishingfield vacuum expectation value generated by radiative corrections is an example ofthe dimensional transmutation phenomenon. This can be seen more clearly if wechoose M = 〈ϕ〉. Then, from (11.27)

λ(〈ϕ〉) = 113 bg4(〈ϕ〉) (11.28)

and we recall that M = 〈ϕ〉 corresponds to on-shell renormalization for theshifted fields. We can take dimensionless λ and g or g and dimensionful 〈ϕ〉 asindependent parameters of our theory.

Spontaneous gauge symmetry breaking leads to the physical consequencesdiscussed in Section 11.1: the would-be Goldstone boson combines with the gauge


boson to make a massive vector meson m2(V ) = g2〈ϕ〉2 and the remaining scalarfield, after shifting, corresponds to a scalar meson of mass m2(S) = V ′′(〈ϕ〉) ∼g4〈ϕ〉2.

Physics cannot depend on the arbitrary parameter M but one choice of M can bemore convenient than the others, particularly since we are working in perturbationtheory and have to control higher order corrections. We have already learned thatthe perturbative calculation of 〈ϕ〉 is reliable when the renormalization point M issuch that λ(M) � g2(M) � 1. It is also worth noting that from (11.27)

〈ϕ〉 = M0e11/6 (11.29)

where λ(M0) = 0. Usually, we suppose that λ(M) and g(M) are given at somescale M = M , for example, at the grand unification scale, and we can then userenormalization group arguments to look for a renormalization scale suitable forour calculation (for example, M0). In our renormalization procedure the scalarquartic coupling is defined in terms of the fourth derivative of V . Thus, tostudy λ(M) one should derive the RGE for this fourth derivative (see Coleman &Weinberg (1973) and Gildener (1976)). We know, however, that the function β(λ)

in the one-loop approximation does not depend on the renormalization conventions.Therefore to the lowest order we can use the results of calculations which adoptmomentum space renormalization conventions. The one-loop RGEs for λ and gthen read

16π2 M(dλ/dM) = Aλ2(M)+ B ′λ(M)g2(M)+ Cg4(M)

16π2 M(dg2/dM) = −bg4(M)

}(11.30)

where the dimensionless coefficients A, B ′ and C are given, respectively, by thediagrams shown in Fig. 11.2. To solve (11.30) we introduce α = λ/g2. Then weget (t = ln M)

16π2

g2(t)

dα

dt= −bg2(t)

dα

dg2= Aα2(t)+ Bα(t)+ C

= A(α − α1)(α − α2), α1 < α2 (11.31)

where B = B ′ + b. For real roots there is a UV stable fixed point at α = α1 < α2

and an IR stable fixed point at α = α2. Thus, if 0 < α(M) = λ(M)/g2(M) < α1

we expect the RGEs (11.31) to have a solution α(M) = 0 for M �= 0. A directintegration gives

exp

(−b

∫ α(M)

0

dα

Aα2 + Bα + C

)= g2(M)

g2(M)= 1 + 1

16π2g2(M)b ln

M

M(11.32)


Fig. 11.2.

or, more explicitly, α(M) = 0 for

M = M exp

[− 16π2

g2(M)F(α(M))

](11.33)

where

F(α(M)) = 1

b

[1 − exp

(−b

∫ α(M)

0

dα

Aα2 + Bα + C

)](11.34)

If α(M) is of the order O(1) (i.e. if λ(M) ∼ g2(M)) then F(α(M)) is also of orderunity and

M

M= exp

[− O(1)

g2(M)

](11.35)

Thus, using (11.29) we conclude that for a reasonably small value of g2(M),the dimensional transmutation can generate a scale M of spontaneous symmetrybreaking which is enormously smaller than the scale M at which as we supposeα(M) is given. This small ratio of mass scales is not put into the theory by handbut arises automatically from the assumption of a vanishing scalar mass. Thusin order for this to happen some massless scalars which escape getting massesof order M must be available in our theory. For instance, if M is the grandunification scale they must remain massless after the spontaneous breakdown ofthe grand unification group. Such a situation can perhaps arise naturally fromsupersymmetries which are unbroken at M . The mechanism described here verymuch resembles the generation of the scale � at which QCD coupling becomeslarge (see (7.53a)). It can also be generalized to the case of several quartic scalarcouplings present in the theory (Gildener & Weinberg 1976).

The generation of a very small ratio of mass scales by radiative corrections andthe dimensional transmutation phenomenon is to be contrasted with making M/Mvery small in the tree approximation. The latter is highly ‘unnatural’. To be morespecific, we have in mind the following: consider a gauge group G which we

11.3 Dynamical breaking of gauge symmetries 379

want to be spontaneously broken to G1 at some scale M , and G1 to be broken toG2 at M with M/M very small. It turns out that one can obtain such a patternof spontaneous symmetry breaking in the tree approximation by a proper choice ofthe scalar potential, but only at the expense of tuning the parameters of the potentialto an accuracy much higher than the neglected one-loop corrections. This is calledthe gauge hierarchy problem.

The formalism presented in this section can be extended to non-abelian gaugetheories with scalars and also fermions (see Coleman & Weinberg (1973) andProblem 11.3).

11.3 Dynamical breaking of gauge symmetries and vacuum alignment

Dynamical breaking of gauge symmetry

We consider now the possibility of spontaneous gauge symmetry breaking in theabsence of elementary scalar fields. The mechanism presented here is often termeddynamical gauge symmetry breaking. Our discussion in this section is closelyconnected with that in Section 10.3.

Imagine a spontaneously broken chiral global symmetry G of some strong gaugeinteractions with gauge group GH (we shall call it the hypercolour group) also tobe broken explicitly by weak interactions with a gauge symmetry group GW; someof the chiral currents of G couple to gauge bosons of GW. By assumption, it is agood approximation to treat GW interaction to the lowest order (see (10.64)). InSection 10.3 we have learned how to find the preferred vacuum which minimizesthe vacuum energy in the presence of the perturbation GW. Assuming the positivitycondition for certain current matrix elements the correct vacuum corresponds to theminimal value of the quantity (see (10.75))∑

α

Tr[XJ αU

XJ αU ] (11.36)

built of the transformed currents projected onto the subspace of the brokengenerators in the original vacuum; we recall that J α is the linear combination of theG generators corresponding to the GW current gα jαµ(x) and XJ α is that part of J α

which contains only broken generators Xα. Equivalently, the correct vacuum mustsatisfy Dashen’s conditions (10.59) and (10.60). Assume that the correct vacuumhas been found: it gives a specific orientation of the spontaneously unbrokensubgroup H in the G, a set of broken generators Xα and the correspondingGoldstone bosons, some of which get masses induced by the perturbation GW. Ingeneral one might also single out another subgroup of G: the maximal subgroup Sof elements of G which commute with all the generators of GW. Then the couplingof gauge bosons to the currents of G breaks G explicitly to GW × S and we get


Fig. 11.3.

Fig. 11.4.

the picture of the G symmetry breaking shown in Fig. 11.3 where the orientationof different subgroups is illustrated in Fig. 11.4.

In the zeroth order in the GW couplings there are Goldstone bosons associatedwith each spontaneously broken generator (regions I + II + III). In presence ofthe GW interactions the Goldstone bosons split into three different groups whichare easy to classify. The GW × S symmetry remains the exact symmetry of thelagrangian and therefore the Goldstone bosons in regions I and II remain masslessin any order in the GW interactions, as can be proved following the arguments ofSection 9.6. The generators in region III do not correspond to exact symmetriesof the full lagrangian and the corresponding Goldstone bosons acquire, in general,masses which in the first order in GW are given by (10.55) or (10.77) and (10.78).From the calculation of the π+–π0 mass difference we can see that the massesare of the order g�(mρ ∼ �), where g is the coupling constant of the weakinteractions and � is the confinement scale of the hypercolour interactions. TheGoldstone bosons in regions I and II still split into two groups: clearly, those inregion II remain in the physical spectrum; however, those in region I are absorbedby the GW gauge bosons which get masses through the Higgs mechanism. Onethen talks about the dynamical breaking of the GW gauge symmetry. Let us discussthe mass generation for the GW gauge bosons somewhat more explicitly. TheGoldstone bosons in region I couple to the GW currents jαµ

〈0| jαµ(0)|πa〉 = ipµ f αa (11.37)


and therefore the gauge-invariant vacuum polarization tensor �αβµν defined by

i�αβµν(p2) = gαgβ

∫d4x exp(ipx)〈0|T jαµ(x) jβν (0)|0〉

= i(gµν p2 − pµ pν)�αβ(p2) (11.38)

has a pole at p2 = 0 (see also Section 11.1)

limp2→0

�αβµν(p2) = −(gµν p2 − pµ pν)

∑a

gαgβ f αa f βa/p2 (11.39)

It is important to remember the role of gauge invariance in writing (11.39): theGoldstone bosons contribute only to the pµ pν term of the polarization tensor; thegµν term must be present by gauge invariance and we do not need to specify itsorigin more explicitly. The full gauge boson propagator αβ

µν reads†

αβµν = −iDµν[δαβ −�αβ(p2)] = −iDµν(1 +�)−1αβ + higher order terms

(11.40)

We are working in the Landau gauge and −iDµνδαβ is the free propagator in this

gauge. Thus, the residue (11.39) gives the vector boson mass matrix

m2αβ =

∑a

gαgβ fαa fβa

The Goldstone boson decay constants f αa can be expressed in terms of theappropriate traces. Under assumption (10.70) we have

〈0| j aµ(0)|πb〉 = ipµ fπδ

ab = ipµ fπ Tr[XaXb] (11.41)

where j aµ is a G current and consequently

m2αβ = f 2

π

∑a

Tr[ J α Xa] Tr[ J β Xa] = f 2π Tr[XJ α XJ β] (11.42)

(where no sum has been taken over α and β). This result has been obtained for thevacuum state |0〉. For any other vacuum state related by the unitary transformationU |0〉 we find

m2αβ = f 2

π Tr[XJ αU

XJ β

U ] (11.43)

Comparing with (10.75) we see that

E(�) ∼ Tr m2 (11.44)

The conclusion is that the energetically preferred vacuum alignment and therelative orientation of the groups GW and H in the presence of the perturbation

† αβµν = (−iDµνδ

αβ)+ (−iDµλδαβ′ (−i�λρ,β′γ ′ )(−iDρνδ

γ ′β)).


GW are such that the trace of the vector boson mass matrix is the minimal one (thegauge symmetry is broken as little as possible). The weak interactions determinetheir own pattern of symmetry breaking by the choice of the vacuum orientation.We note also that the vector boson mass matrix obeys, in general, certain relationswhich are consequences of H invariance.

The above ideas may be relevant for the gauge symmetry breaking in theGlashow–Salam–Weinberg theory of the weak interactions (see, for instance, Fahri& Susskind (1981)), however, no convincing model has been constructed yet. Wewill illustrate them with two simple examples.

ExamplesConsider a strong hypercolour gauge group GH with 2N left-handed fermions and2N right-handed fermions transforming as the same representation of GH. Thetheory has G = SUL(2N )× SUR(2N ) chiral symmetry which, we assume, breaksdown spontaneously to H = SU (2N ). The embedding of H in G is characterizedby a fermion condensate

〈0|�Li�R j |0〉 = �i j (11.45)

As in Section 9.6, assuming �i j = vδi j , the chiral group is broken to SUL+R(2N ).Any equivalent orientation of the unbroken SU (2N ) corresponds to a condensateobtained from (11.45) by a unitary transformation

UR�U †L = vURU †

L (11.46)

(remember that U †�U = UR�U †L;U is an unitary operator, UL(R) are unitary

matrices), i.e. it is specified by a unimodular unitary matrix URU †L. We take the

weak gauge group to be GW = SUL(2) × U (1) with fermions in N left-handedGW doublets and 2N right-handed singlets, and with the U (1) charge assignmentsas follows (see Chapter 12): (

Ui

Di

)L

UiR DiR

Y = q q + 12 q − 1

2

Consider first the case N = 1. Denoting the gauge bosons of the Weinberg–Salam model by W α

µ , Bµ the weak coupling is

L = −∑α

gW αµ�Lγ

µT α�L − g′Bµ(�RγµT 3�R + q�γ µ�) (11.47)

where T α is an SU (2) generator normalized to Tr[T αT β] = 12δ

αβ and q is the meanelectric charge of the doublet. Assuming the condensate (11.45) and expressing the


GW currents jαµ in terms of the G currents corresponding to the unbroken subgroup(vector currents) and to the broken generators (axial currents) jL = 1

2(V−A), jR =12(V +A) one gets in the (�L, �R) basis the following expression:

J α = g

(T α

0

)+ g′

(q

q + T 3

)=[

12 g

(T α

T α

)+ 1

2 g′(

2q + T 3

2q + T 3

)]+[

12 g

(T α

−T α

)+ 1

2 g′(−T 3

T 3

)](11.48)

Inserting (11.48) into (11.42) gives the familiar vector boson matrix

W1 W2 W3 B

W114 g2 f 2

π

W214 g2 f 2

π

W314 g2 f 2

π − 14 gg′ f 2

π

B − 14 gg′ f 2

π14 g′2 f 2

π

After diagonalization one gets

mW1,2 = 12 g fπ , mZ = 1

2 fπ(g′2 + g2)1/2 = mW/ cos�W

mγ = 0

}(11.49)

where �W, the Weinberg angle, is given by

cos�W = g

(g′2 + g2)1/2

and

γ = cos�W B + sin�WW3

Z = − sin�W B + cos�WW3

In this model all the Goldstone bosons have been used to give masses to the weakgauge bosons. There is no problem of vacuum alignment since all the equivalentvacua (11.46) can be transformed into (11.45) by the gauge transformations in theweak sector.

For N > 1 left-handed doublets with the same U (1) charge assignment thesituation is more complex. If we introduce the 4N -plet � = (U1LD1L . . .

DNLDNLU1RD1R . . .UNRDNR) then the Salam–Weinberg coupling is

L = −gW αµ�γ µ

(T α ⊗ 1l

0

)� − g′Bµ�γ µ

(q

q + T 3 ⊗ 1l

)� (11.50)

where the 2N × 2N block T α ⊗ 1l is a direct product of the 2 × 2 weak isospinand the N × N matrix on the space of N doublets. The weak coupling with


the gauge symmetry GW = SUL(2) × U (1) breaks the original chiral symmetryG = SUL(2N ) × SUR(2N ) of the strong interaction sector into G → GW × Swhere S = SUL(N ) × SUR(N ) × SUR(N ) with the first SU (N ) correspondingto horizontal transformations among N left-handed doublets, and the second (thethird) SU (N ) corresponding to transformations among N right-handed singletsUiR(DiR);UiR and DiR have different charges and therefore the original SUR(2N )

is broken into SUR(N ) × SUR(N ). The spontaneous breakdown of G intoSUL+R(2N ) induced by (11.46) leaves the subgroup SUL+R(N ) of S unbroken.The spontaneous breakdown of the original SU (2N ) × SU (2N ) chiral symmetryproduces 4N 2 − 1 Goldstone bosons. Three of them are used to give masses to thegauge bosons. Again, there is no vacuum alignment problem. The breaking of Sinto SU (N ) which remains the exact symmetry of the full theory leaves 2(N 2 − 1)exactly massless Goldstone bosons. They are electrically neutral because theycorrespond to generators of transformations among fermions with the same charge.There remain 2(N 2 − 1) charged Goldstone bosons which do not correspond togenerators of exact symmetries of the full lagrangian and they should, therefore,receive masses from the GW interactions. Splitting the weak currents into theunbroken YJ α and the broken parts XJ α as in (11.48)

YJ α ≡ 12 g

(T α ⊗ 1l

T α ⊗ 1l

)+ 1

2 g′(

2q + T 3 ⊗ 1l2q + T 3 ⊗ 1l

)XJ α ≡ − 1

2 g

(−T α ⊗ 1lT α ⊗ 1l

)+ 1

2 g′(−T 3 ⊗ 1l

T 3 ⊗ 1l

) (11.51)

it is evident from (10.79) that the Goldstone bosons receive no mass in the firstorder in GW. The broken generators Xa are of the block-diagonal form(−A

A

)where the As are 4N 2− 1 matrices which can be easily constructed for any N , andtherefore for YJ and XJ given by (11.51) the two contributions to the trace in (10.79)cancel each other. Actually, one can see that if GW commutes with one of the chiralgroups SUL(2N ) or SUR(2N ) then there will be pseudo-Goldstone bosons whichremain massless to the lowest order in GW. Indeed, one sees directly from (10.66),using H invariance as in (10.69), that the �-dependent (U -dependent) part of thevacuum energy E(�) relevant for the pseudo-Goldstone boson masses (10.79)comes from the term

E(�) = − 12

∫d4x µν(x)

∑α

g2α〈0|T U †

L jαµL(x)ULU †R jανR(0)UR|0〉 (11.52)

whereas jαµL jανL and jαµR jανR terms contribute only the �-independent constants.


Fig. 11.5.

In our example the U (1) current also does not contribute to E(�) (althoughit has both the left-handed and the right-handed parts) because the left-handedU (1) current is an SUL(2N ) singlet. The U (1) contribution is therefore SUL(2N )-invariant and also SUL+R(2N )-invariant because the vacuum is, so it is SUL(2N )×SUR(2N )-invariant and hence �-independent. The charged pseudo-Goldstonebosons receive masses in the next order in GW due to the mass splitting of the γ

and Z which has to be taken into account in that order, for example, as in Fig. 11.5.In the model discussed above there remain massless neutral Goldstone bosons

in the physical spectrum because the SU (N ) group of transformations betweendoublets remains the exact symmetry of the theory. This can be avoided bygiving the left-handed doublets different U (1) charge. Such a model is veryinstructive because then the U (1) boson exchange contributes the �-dependentterm to the E(�) and the pattern of the GW breakdown depends on the U (1)charge assignments.

To be specific we consider a model with four left-handed and four right-handedfermions with the GW representation content given by(

U1

D1

)L

(U2

D2

)L

U1R D1R U2R D2R

Y = , − , ( + 12), ( − 1

2), (− + 12), (− − 1

2)

(11.53)

Similarly to (11.50) the GW coupling in the �L, �R basis (� =(U1LD1LU2LD2LU1RD1RU2RD2R)) is

L = −gW αµ�γ µ

(T α ⊗ 1l

0

)� − g′Bµ�γ µ

[(0

T 3 ⊗ 1l

)

+

(1l ⊗ T 3

1l ⊗ T 3

)]� (11.54)

where in the product A ⊗ B the first matrix is in the 2 × 2 weak space and thesecond one is on the space of doublets. Let us first assume that the SU (4)× SU (4)symmetry of the GH lagrangian is broken by the

〈0|ULiUR j |0〉 = 〈0|DLiDR j |0〉 = vδi j (11.55)


condensates. This defines the vacuum and the breaking SUL(4) × SUR(4) →SUL+R(4) in the basis (11.53). We can now split the GW currents into unbrokenand broken parts in the vacuum (11.55). It is clear from our previous discussionthat only the U (1) current is interesting to us; as before the contribution of theSU (2) current to the two terms under the trace in (10.79) cancels. For the U (1)current we have

J = 12 g′

(T 3 ⊗ 1l + 2 1l ⊗ T 3

T 3 ⊗ 1l + 2 1l ⊗ T 3

)+ 1

2 g′(−T 3 ⊗ 1l

T 3 ⊗ 1l

)(11.56)

There are 15 Goldstone bosons corresponding to the broken generators

X = 1

2

(−AA

)where A represents T a ⊗ 1l, 2T a ⊗ T 3, 1l ⊗ T 3 and eight matrices of the form, forexample,

1√2

01 00 0

1 00 0

0

or1√2

0−i 00 0

i 00 0

0

(11.57)

linking one of (U1,D1) with one of (U2,D2). A look at (10.79) and (11.56) tellsus that Goldstone bosons corresponding to broken generators T a ⊗ 1l, T a ⊗ T 3

and 1l ⊗ T 3 receive no mass in the first order in GW. An explicit calculationbased on (10.79), (11.56) and (11.57) gives the following masses for the remainingpseudo-Goldstone bosons which are denoted by the flavours linked by the corre-sponding generators: non-diagonal terms of the mass matrix vanish and bosonscorresponding to the two generators like (11.57) are degenerate in mass and

m2U1U2

= m2D1D2

∼ g′2 2 M2

m2U1D2

∼ g′2 ( + 1)M2

m2D1U2

∼ g′2 ( − 1)M2

(11.58)

where M2 is the integral given by (10.78). Assuming that the integral M2 isdominated by the lowest-lying vector mesons of the correct quantum numbers wefind, as for the π+–π0 mass difference in Section 10.4, M2 > 0. Therefore, we findan interesting result: for | | > 1 Dashen’s condition is satisfied by masses (11.58)and the vacuum (11.55) is the preferred vacuum in the presence of perturbationGW. However, for the U (1) charge assignments for doublets such that | | < 1the vacuum (11.55) turns out to be unstable. We must, therefore, find the correct


vacuum in the set of vacua obtained from (11.55) by all unitary transformations Uof SU (4)× SU (4). According to (11.36) the right transformation is the one whichminimizes the Tr XJ 2

U or maximizes the Tr YJ 2U of the projection of the transformed

current on the subspace of the original broken (Xa) and unbroken (Y a) generators,respectively. Before we find this transformation let us notice that in the originalbasis (11.54) the term in the lagrangian proportional to is SUL+R(4)-invariant.It is then clear that for | | > 1 that basis is the right one: the L given by (11.54)is ‘as far as possible’ SUL+R(4)-invariant. However, for | | < 1 one can imaginethat a transformation U † JU , which breaks the SUL+R(4)-invariance of the term,simultaneously transforms the other term of the U (1) current to such a form thatthe full transformed current has ‘better’ SUL+R(4) symmetry than the original one.The right transformation can be found by noting that the transformed U (1) current,split into unbroken and broken parts, is as follows:

U †

−

−

1 +

−1 +

1 −

−1 −

U

=(

AA

)+(

B−B

)where the matrix on the l.h.s. is the U (1) current in the original basis (11.54).Since the trace of B2 should be the minimal one, the transformed current shouldhave both 4 × 4 blocks as similar as possible (A + B ≈ A − B ≈ A). Hence, for| | < 1 the right transformation is the one which gives

U † JU =

−

−

1 +

1 −

−1 +

−1 −


From (11.54) one sees that this is the transformation D1R ↔ U2R. The new vacuumis then defined by the condensate

UR�U †L = v

U1L D1L U2L D2L1 0 0 00 0 1 00 1 0 00 0 0 1

U1R

D1R

U2R

D2R

so the electrically charged composite field U2D1 acquires a non-zero vacuumexpectation value. Thus, the U (1)em subgroup of the gauge group SU (2) × U (1)is broken and the photon has a mass when | | < 1. Of course, one can easily finda set of broken and unbroken generators corresponding to the new vacuum as wellas all the gauge boson masses and pseudo-Goldstone boson masses.

An interesting conclusion of the above exercise is that the coupling of a weakgauge group to the chiral currents of a spontaneously broken chiral symmetryof strong interactions may, for dynamical reasons, lead to different theoriesdepending on the specific assignment of the chiral multiplets to the weak grouprepresentations.

Problems

11.1 Derive the remaining Feynman rules for the U (1) theory of Section 11.1 in the Rξ

gauge.

11.2 Check by explicit calculation that at the tree level the unphysical ξ -dependent polescancel in the S-matrix for the process ϕ1 + ϕ2 → A1 + A2 where ϕ and A arethe physical scalar particle and the massive vector meson of the U (1) theory inSection 11.1, respectively.

11.3 Extend the discussion of Section 11.2 to non-abelian gauge theories includingfermions. Calculate the Higgs particle (physical scalar) mass in terms of the W andZ masses in the SU (2)×U (1) electroweak theory with one massless scalar complexdoublet and taking fermions as massless.

12

Standard electroweak theory

The basic concepts of the electroweak theory (Glashow 1961, Weinberg 1967b,Salam 1968) have been outlined in the Introduction. The underlying symmetrygroup SUL(2) × UY(1) emerges from study of the charged weak currents, sup-plemented by the idea of minimal unification with electromagnetic forces. Thesymmetry is assumed to be local gauge symmetry, in accordance with generalremarks in the Introduction and as required by unification with electromagnetism.Thus, we have the following gauge vector fields:† W a

µ for SUL(2) and Bµ forUY(1). The matter fields consist of quarks and leptons forming three generations.In terms of the left-handed Weyl fields each generation consists of:

lA(1, 2,− 1

2

)ec

A (1, 1,+1)

qA(3, 2,+ 1

6

)dc

A

(3�, 1,+ 1

3

)uc

A

(3�, 1,− 2

3

)

(12.1)

where A = 1, 2, 3 is the generation index. (Any reader unfamiliar with thetwo-component spinor notation should consult Chapter 1 and Appendix A.) Thefirst and second entries show the transformation properties under the colour SU (3)and the weak SUL(2) groups, respectively. The last entry shows the hyperchargeof each field. We recall that the hypercharge is defined as Y = Q − T 3, where Qis the electric charge of the field, and T 3 is the third component of its weak isospinSUL(2). At this stage the superscript ‘c’ does not have the same meaning as inChapter 1. It is just used to denote the left-handed Weyl fields which are singlets ofSUL(2). It will acquire the proper meaning of charge conjugation (as in Chapter 1)after the electroweak symmetry breaking to the electromagnetic U (1). Thehermitean conjugates of the fields carrying it will then combine with appropriate

† The SU (3) gauge group of QCD was extensively discussed in Chapter 8. The complete SUc(3)× SUL(2)×UY(1) theory is called the Standard Model of elementary interactions.

389

390 12 Standard electroweak theory

components of the doublets to form Dirac fields, with charge conjugation actingonly on their electric and colour (i.e. conserved) charges. Writing explicitly theSUL(2) doublet components and using the conventional terminology we have

l1 ≡(νe

e

), l2 ≡

(νµ

µ

), l3 ≡

(ντ

τ

)ec

1 ≡ ec, ec2 ≡ µc, ec

3 ≡ τ c

for leptons and

q1 ≡(

ud

), q2 ≡

(cs

), q3 ≡

(tb

)uc

1 ≡ uc, uc2 ≡ cc, uc

3 ≡ t c

dc1 ≡ dc, dc

2 ≡ sc, dc3 ≡ bc

for quarks.We can also introduce the four-component chiral fermion fields, as in (1.72):

(� ilA)L ≡

(li

A

0

), (�eA)R ≡

(0ec

A

),

(� iqA)L ≡

(qi

A

0

), (�dA)R ≡

(0

dcA

), (�u A)R ≡

(0

ucA

) (12.2)

where i = 1, 2. We recall that the unbarred (barred) two-component spinors carrylower (upper) Lorentz indices. The left-(right-)handed chiral fields �L(R) ≡ 1

2(1 ∓γ5)�L(R) are doublets (singlets) under SUL(2) and the electric and colour chargesof the left- and right-handed fields are pairwise the same. We shall also use thenotation (�1

lA)L ≡ (�νA)L.

The gauge symmetry SUL(2) × UY(1) must be spontaneously broken to theelectromagnetic UEM(1) or, more precisely, the so-called Higgs mechanism (dis-cussed in Section 11.1) must be invoked. To this end one introduces a doubletof scalar fields H , called the Higgs doublet, with the quantum numbers (1, 2, 1

2)

with respect to the full SU (3) × SUL(2) × UY(1) gauge group. One Higgsdoublet is the minimal set of scalar fields which allows for the required symmetrybreaking. Since the presence of more doublets (for constraints on the presenceof other representations of scalars see Problem 12.1) has no particular theoreticalmotivation (apart from the supersymmetric models which require an even numberof Higgs doublets), the one-doublet theory is what is called the Standard Model.The Higgs mechanism is the origin of the masses of the W± and Z 0 vectorbosons (which are linear combinations of the original W a

µ and Bµ fields) and ofthe masses of elementary fermions, which are strictly massless in the symmetrylimit. It does not destroy the renormalizability of the theory (’t Hooft 1971a, b).

12.1 The lagrangian 391

Moreover, interestingly enough, Higgs doublets and only Higgs doublets can haverenormalizable Yukawa couplings to the left- and right-handed chiral matter fields.Therefore, after symmetry breaking, they can generate the Dirac mass terms forthose particles.

As discussed in Chapter 9, the pattern SU (2) × U (1) → U ′(1) is unique if theHiggs mechanism is generated by vacuum expectation values of the componentsof a complex scalar SU (2) doublet field H(x). By proper redefinition of thegroup generator basis the remaining U ′(1) can always be interpreted as theelectromagnetic UEM(1). With the basis fixed by the charge assignment (12.1)the desired pattern is obtained with

H(x) =(

H+(x)H 0(x)

)(12.3)

where

〈0|H(x)|0〉 = 1√2

(0v

)(12.4)

and the components H+ and H 0 are complex fields. It may be useful to stressthat, according to (12.4), the vacuum carries quantum numbers of the componentH 0. This is different from quantum vacuum fluctuations which are singlets withrespect to all symmetries, and are eliminated by normal ordering (a redefinition ofthe vacuum energy) which is understood also to be present in (12.4). As discussedin Chapters 1 and 9, spontaneously broken symmetries can be intuitively depictedas the property of the vacuum which is a big reservoir of the scalars H 0.

The electroweak theory has been confirmed experimentally with the precision ofone per mille, mainly by the LEP experiments, as the correct theory of elementaryinteractions for the energy range up to O(100 GeV). In particular, the quantumone- and partly two-loop corrections to the tree level predictions have been tested(see later). However, the Higgs scalar has not yet (1999) been discovered, andis extensively searched for in present and planned experiments. From the pointof view of the electroweak theory considered in the range up to O(100 GeV),the Higgs boson mass range can extend up to O(1 TeV). It can be elementaryor composite or even only the effective description of a strongly interacting (atthe scale O(1 TeV)) gauge boson sector (Section 11.3). Thus, full experimentalclarification of the Higgs sector is expected to provide an important bridge tophysics beyond the Standard Model.

12.1 The lagrangian

With the ideas and the formalism introduced so far in this book we are wellequipped to formulate the electroweak theory (Glashow 1961, Weinberg 1967b,


Salam 1968, ’t Hooft 1971a, b) as a gauge theory. The full classical gauge-invariantlagrangian consists of several parts:

LSM = Lgauge + Lmatter + LHiggs + LYukawa + Lg.fix + Lghosts (12.5)

The gauge boson part has the standard form:

Lgauge = − 14 W a

µνW aµν − 14 Bµν Bµν (12.6)

where W aµν = ∂µW a

ν − ∂νW aµ − gεabcW b

µW cν and Bµν = ∂µBν − ∂ν Bµ are the field

strengths of the W aµ and Bµ gauge fields and εabc are the SUL(2) group structure

constants. Its kinetic and interaction parts are, respectively,

Lkingauge = 1

2 W aµ

(gµν∂2 − ∂µ∂ν

)W a

ν + 12 Bµ

(gµν∂2 − ∂µ∂ν

)Bν

Lintgauge = gεabc

(∂µW aν

)W b

µW cν − 1

4 g2(W aµW bνW a

µW bν − W aµW aνW b

µW bν

)The matter field lagrangian in the two-component spinor notation reads

Lmatter = il Aσµ

(∂µ + igW a

µT a − i

2g′Bµ

)lA

+ iecAσ

µ(∂µ + ig′Bµ

)ec

A

+ iqAσµ

(∂µ + igW a

µT a + i

6g′Bµ

)qA

+ idcAσ

µ

(∂µ + i

3g′Bµ

)dc

A

+ iucAσ

µ

(∂µ − i

2

3g′Bµ

)uc

A (12.7)

where T a are the SUL(2) generators in the fundamental representation and g, g′ arethe two coupling constants associated with SUL(2) and UY(1) groups, respectively.The generation index A is summed from 1 to 3. In terms of four-componentchiral fermions defined in (12.2) the lagrangian (12.7) can be rewritten in the morefamiliar form

Lmatter = i(�lA)Lγµ

(∂µ + igW a

µT a − i

2g′Bµ

)(�lA)L

+ i(�eA)Rγµ(∂µ − ig′Bµ

)(�eA)R

+ i(�qA)Lγµ

(∂µ + igW a

µT a + i

6g′Bµ

)(�qA)L

+ i(�dA)Rγµ

(∂µ − i

3g′Bµ

)(�dA)R

+ i(�u A)Rγµ

(∂µ + i

2

3g′Bµ

)(�u A)R (12.8)

12.1 The lagrangian 393

(Note the change of signs of the U (1) charges of the SUL(2) singlets in (12.8)compared to (12.7).)

The lagrangian for the Higgs field has the form

LHiggs = H †

(←∂µ − igT aW a

µ −i

2g′Bµ

)(→∂µ + igT aW a

µ +i

2g′Bµ

)H

− m2 H † H − λ

2

(H † H

)2(12.9)

Interactions of the Higgs doublet with the matter fields are described by theYukawa part of the full lagrangian (12.5):

LYukawa = −Y B Al H �

i li AecB − Y B A

d H �i qi Adc

B − Y B Au εi j Hi q j Auc

B + h.c. (12.10)

where ε12 = −ε21 = −1, Y AB are 3 × 3 complex Yukawa coupling matrices andsummation over the SUL(2) indices i, j and over the fermion generation indicesA and B is understood. We have also used the fact that the two-dimensionalrepresentation of SU (2) is real (see Appendix E) and iτ2 H transforms as H �, i.e.as 2� of SUL(2). Therefore, (iτ2 H) j q j = εi j Hi q j is also an invariant of SUL(2).†Using the four-component chiral fields this can be rewritten as

LYukawa = −Y B Al H �

i (�eB )R(�ilA)L − Y B A

d H �i (�dB )R(�

iqA)L

− Y B Au εi j Hi (�u B )R(�

jqA)L + h.c. (12.11)

When m2 < 0 is chosen in (12.9) the Higgs doublet acquires the vacuumexpectation value. Indeed, the minimum of the potential is for

〈H † H〉 = −m2

λ≡ v2

2(12.12)

By SUL(2) rotation we can always redefine the vacuum so that only the vacuumexpectation value of the lower component of the Higgs doublet is non-vanishing.The SUL(2)×UY(1) symmetry is then broken down to U ′(1) (which is identifiedwith the electromagnetic U (1) with Q = T 3 + Y ) because

T a

0v√2

�= 0, for a = 1, 2, 3, (T 3 + Y )

0v√2

= 0 (12.13)

where Y is the hypercharge generator.

† In the lagrangian (12.10) we have not included the Yukawa couplings for neutrinos since our list (12.2) ofphysical fields does not contain the right-handed neutrinos. Note, however, that the right-handed neutrinos, ifthey exist, are sterile (T3 = Y = 0) with respect to the electroweak interactions and their inclusion would bea no-cost extension of the theory. The neutrino Yukawa terms and explicit Majorana mass terms for the right-handed neutrinos can then be added to the electroweak lagrangian without destroying the renormalizability ofthe theory. Such an extension may be necessary in view of the accumulating evidence for neutrino masses.


The gauge-fixing term and the Fadeev–Popov ghost term of the lagrangian (12.5)will be discussed later. First, we shall discuss the predictions of the theory whichfollow from the underlying gauge symmetry and the Higgs mechanism, i.e. fromthe tree level structure of the lagrangian.

12.2 Electroweak currents and physical gauge boson fields

The gauge boson couplings to matter follow from Lmatter and have the form

Lmatter ⊃ −g J aµW aµ − g′ Jµ

Y Bµ (12.14)

where (see (12.8))

J aµ =∑

f

�f

L γ µT a�f

L (12.15)

Jµ

Y =∑

f

�f

L γ µY f �f

L +∑

f

�f

Rγ µY f �f

R (12.16)

The SUL(2) current has, of course, explicit V –A structure. Note that the U (1)current is chiral, too. Both left- and right-handed fermions contribute to it, butwith different U (1) charges. This is a potential source of chiral anomalies (seeChapter 13).

It follows from the unification hypothesis that the photon field Aµ is a com-bination of the Yang–Mills fields W 3

µ and Bµ. This combination can be easilyfound either from the requirement that the photon does not couple to the neutralcomponent of the Higgs field, or from the requirement that its coupling to fermionsis vector-like with the strength given by their electric charges. We find

Aµ = 1

(g2 + g′2)1/2(gBµ + g′W 3

µ) ≡ cos θW Bµ + sin θWW 3µ (12.17)

where the Weinberg angle is defined as

sin2 θW ≡ g′2

g2 + g′2, tan θW ≡ g′

g(12.18)

and the electric charge is (e > 0)

e ≡ g′g(g2 + g′2)1/2

= g sin θW (12.19)

The orthogonal combination defines another electrically neutral field of the theory:

Z0µ =

1

(g2 + g′2)1/2(gW 3

µ − g′Bµ) ≡ cos θWW 3µ − sin θW Bµ (12.20)

12.2 Electroweak currents 395

Its couplings to fermions then follow from (12.14):

g

cos θWJµ

Z Z0µ =

g

cos θW(J 3µ − sin2 θW Jµ

EM)Z0µ (12.21)

with

Jµ

EM =∑

f

�f

L γ µQ f �f

L +∑

f

�f

Rγ µQ f �f

R (12.22)

The mixing angle θW measures the departure of the weak neutral current Jµ

Z frompure V –A structure! It is measured experimentally, for example, in atomic parityviolation experiments (for other measurements of sin2 θW – see Section 12.4), andtogether with the electric charge can be used as independent input parameter fixingthe couplings g and g′. It is worth stressing that the departure in the neutral currentfrom pure V –A Lorentz structure is the basic prediction of the unified theory.Moreover, the value of the Weinberg angle has a priori nothing to do with thevector boson masses, but is just a measure of this departure. Such a relation is,however, obtained after the W± and Z0 bosons are given masses by the Higgsmechanism, and it depends crucially on the SUL(2)×UY(1) representation contentof the Higgs fields.

The physical gauge and Higgs boson spectrum of the electroweak theory is mosteasily seen in the so-called unitary gauge introduced in Chapter 11. Using the‘angular’ variables for the Higgs field

H(x) = 1√2

exp

[iηa(x)

vT a

](0

v + h0(x)

)(12.23)

and choosing the gauge rotations �aL(x) = −ηa(x)/v, we again see from (12.9)

that the minimum of the scalar potential occurs for v2 = −2m2/λ. Furthermore,three of the four degrees of freedom present in the Higgs doublet H(x) areexchanged for the longitudinal components of the gauge bosons W±

µ ≡ 1√2(W

1µ ∓

iW 2µ) and Z0, which obtain the masses

M2W = 1

4 g2v2, M2Z = 1

4(g2 + g′2)v2 (12.24)

The photon remains massless, in accord with the fact that the U (1)EM gaugesymmetry remains exact. The W± and Z0 boson masses are related by

MW

MZ= cos θW (12.25)

(actually the same result holds with any number of Higgs doublets; the reader isinvited to check this by explicit calculation). Also from (12.9) we see that the fourthdegree of freedom originally present in the field H(x) remains in the physicalspectrum as an electrically neutral massive Higgs scalar with the mass M2

h = λv2.


Since in the unitary gauge the theory is not manifestly renormalizable, inthe following we choose to work with the covariant gauge-fixing condition. Inthis class of gauges, the unphysical Higgs fields (the three Goldstone bosons‘eaten up’ in the unitary gauge by the longitudinal components of the gaugeboson fields) remain in the lagrangian (and in the Feynman rules). However,as discussed in Section 11.1, the unphysical poles of the Green’s functions donot, of course, contribute to the S-matrix elements. We recall that S-matrixelements are obtained from the Green’s functions by removing external lines,setting external momenta on the mass-shell, and contracting tensor indices withappropriate physical polarization vectors.

We complete this section by discussing some aspects of the lagrangian incovariant (renormalizable) gauges. Before fixing the gauge we write† in general

H =(

G+1√2(v + h0 + iG0)

)(12.26)

where h0 is assumed to have a vanishing vacuum expectation value. Inserting(12.26) in the scalar potential in (12.9) one finds (with v2 = −2m2/λ fixed asusual in its minimum) that the fields G0 and G± remain in the lagrangian (12.9)and mix with the gauge bosons (v can always be taken as a real parameter):

LHiggs = · · · + i

2gv(W+

µ ∂µG− − W−µ ∂µG+)− 1

2(g2 + g′2)1/2vZ0

µ∂µG0 (12.27)

In the so-called Rξ gauge these terms are, however, cancelled by the appropriatechoice of the Lg.fix part of the lagrangian. In lagrangian (12.9) we also have thecanonical kinetic terms for h0, G0 and G+ fields as well as their self-interactionsand interactions with the gauge fields Aµ, Z 0

µ and W±µ . These terms are straight-

forward to obtain and their derivation is left as an exercise for the reader. Finally,one finds the mass terms for the gauge bosons

LHiggs = · · · + 18 g2v2(W 1µW 1

µ + W 2µW 2µ)+ 1

8v2(gW 3µ − g′Bµ)(gW 3

µ − g′Bµ)

≡ M2W W+µW−

µ + 12 M2

Z Z0µZ0µ (12.28)

The fields G0 and G± are massless at this stage but they will be given masses bythe gauge-fixing term.

The terms (12.27) which mix W± and Z0 vector bosons with the Goldstone

† Compared to (12.3) we have renamed the Goldstone boson fields which in the unitary gauge are ‘eaten up’by the longitudinal components of the physical massive gauge boson fields. Also note that the neutral field issplit into two real fields.

12.2 Electroweak currents 397

fields are cancelled by the following gauge-fixing lagrangian†

Lg.fix = − 1

2ξ

3∑a=0

FaFa (12.29)

with

F0 ≡ ∂µBµ − iξg′(H †Y 〈H〉 − 〈H †〉Y H

)Fa ≡ ∂µW a

µ − iξg(H †T a〈H〉 − 〈H †〉T a H

)(a = 1, 2, 3)

}(12.30)

or, more explicitly,

F0 ≡ ∂µBµ − 1

2ξg′vG0

F3 ≡ ∂µW 3µ +

1

2ξgvG0

F1 ≡ ∂µW 1µ −

i

2√

2ξgv(G− − G+)

F2 ≡ ∂µW 2µ −

1

2√

2ξgv(G− + G+)

Integrating these terms by parts, one gets

Lg.fix = 1

2ξZ0

µ∂µ∂ν Z0

ν +1

2ξAµ∂

µ∂ν Aν + 1

ξW+

µ ∂µ∂νW−ν

− 1

2ξ M2

Z G0G0 − ξ M2W G+G−

+ 1

2(g2 + g′2)1/2vZ0

µ∂µG0 − i

2gv(W+

µ ∂µG− − W−µ ∂µG+) (12.31)

The last two terms indeed cancel the terms in (12.27). The first three terms aregauge-dependent corrections to the kinetic terms for the gauge bosons, and theremaining two terms are mass terms for the unphysical Goldstone bosons.

Finally, the ghost lagrangian can be obtained using the general proceduredescribed in Chapter 3. Performing infinitesimal gauge transformations on thefields in Fa in (12.30):

δY H = −ig′δ�YY H, δL H = −igδ�aLT a H

δY H † = ig′δ�Y H †Y, δL H † = igH †δ�aLT a

δY Bµ = ∂µδ�Y, δYW aµ = 0

δL Bµ = 0, δLW aµ = −gεabcW b

µδ�cL + ∂µδ�

aL

(12.32)

† In general, one can introduce different gauge-fixing parameters ξ for different components of the W a fieldand for the B field. We follow here the simplest approach and introduce only one such parameter.


we compute the corresponding Fadeev–Popov determinants (the variations aretaken always at � = 0):

MYY(x, y) ≡ δF0(x)

δ�Y(y)= [

∂2 + ξg′2(H †Y Y 〈H〉 + 〈H †〉Y Y H

)]δ(4)(x − y)

MacLL(x, y) ≡ δFa(x)

δ�cL(y)

= [∂2 + gcabc←∂

µW b

µ + ξg2(H †T cT a〈H〉

+ 〈H †〉T aT c H)]δ(4)(x − y)

MaLY(x, y) ≡ δFa(x)

δ�Y(y)= ξgg′

(H †Y T a〈H〉 + 〈H †〉T aY H

)δ(4)(x − y)

MaYL(x, y) ≡ δF0(x)

δ�aL(y)

= ξgg′(H †T aY 〈H〉 + 〈H †〉Y T a H

)δ(4)(x − y)

(12.33)

(the symbol←∂µ

means that the derivative acts on the object standing to its left inthe full expression for the ghost action). The ghost action then reads

Sghost =∫

dx∫

dy[ηY(x)MYY(x, y)ηY(y)+ ηa

L(x)MabLL(x, y)ηa

L(y)

+ ηY(x)MaYL(x, y)ηa

L(y)+ ηaL(x)Ma

LY(x, y)ηY(y)]

(12.34)

Since the variations of Fa contain factors δ(4)(x − y), this can be written as anintegral of a local lagrangian density

Sghost =∫

dx Lghost(x) (12.35)

Finally, defining

ηZ = cos θWη3L − sin θWηY,

ηA = sin θWη3L + cos θWηY,

η± = 1√2

(η1

L ∓ iη2L

),

(12.36)

and expressing the functional derivatives (12.33) through the physical fields W±µ ,

Z0µ, Aµ and h0, one arrives at Lghost generating the Feynman rules collected in

Appendix C. We shall work in the so-called ’t Hooft–Feynman gauge with ξ = 1.

12.3 Fermion masses and mixing

The Yukawa terms (12.10) in the electroweak lagrangian couple the left-handedWeyl fermion doublets of SUL(2) with the left-handed Weyl fermion singlets andthe Higgs boson. Clearly, all those fields have well-defined symmetry propertiesunder SUL(2) × UY(1) and we call them to be in an ‘electroweak basis’. With

12.3 Fermion masses and mixing 399

three families of fermions, this basis is defined up to rotations in three-dimensionalflavour space, independently for each category of fields qA, lA, ec

A, ucA, dc

A (A =1, 2, 3). Yukawa couplings are free parameters of the present theory. One mayhope that the future, more fundamental theory will fix those parameters in someelectroweak basis.

A non-zero vacuum expectation value of the Higgs field (12.26) generatesfermion mass terms:

Lmass = − v√2

Y B Al eAec

B −v√2

Y B Ad dAdc

B −v√2

Y B Au u Auc

B + h.c. (12.37)

The physical interpretation of these terms becomes more transparent when werewrite the lagrangian (12.37) in terms of the four-component chiral fields, definedin (12.2):

Lmass = − v√2

Y B Al (�eB )R(�eA)L − v√

2Y B A

d (�dB )R(�dA)L

− v√2

Y B Au (�u B )R(�u A)L + h.c. (12.38)

The mass matrices are:

[me] = v√2

[Ye], [md] = v√2

[Yd], [mu] = v√2

[Yu] (12.39)

In general the Yukawa matrices can be the most general complex 3 × 3 matrices.They can be diagonalized by two unitary transformations (check it): V †[m]U →[m]diagonal with real diagonal entries; the latter follows from the fact that thetransformations V and U can include phase rotations independently for the left-and right-handed fields. Thus, to diagonalize the mass matrices we perform unitarytransformations on the right- and left-handed fields, independently for up and downquarks and charged leptons:

(�u A)L = (UL)AB(�′u B

)L, (�u A)R = (UR)AB(�′u B

)R

(�dA)L = (DL)AB(�′dB)L, (�dA)R = (DR)AB(�

′dB)R

}(12.40)

and analogously for the down quarks and charged leptons eA (i.e. e, µ, τ ). Theprimed fields describe physical particles: after diagonalization we can combinechiral fields into Dirac fields � = �L +�R, which are mass eigenstates.

As mentioned in Section 12.1 one consequence of the lagrangian (12.10) isthat neutrinos remain massless. The accumulating experimental evidence forneutrino masses and oscillations (for a review see, for example, Fisher, Kayser& McFarland (1999)) may call for some extension of the theory. An attractiveand straightforward extension is the inclusion of the right-handed neutrinos or,strictly speaking, of a set of new left-handed Weyl fields νc

L in the list (12.1)


with four-component (�νA)R given by the relation analagous to (12.2). Newrenormalizable terms can then be added to the electroweak lagrangian, namely,neutrino Yukawa terms Y B A

ν εi j Hi l Aj ν

cBL and the right-handed neutrino Majorana

mass matrix M B AνcAL νcB

L + h.c. Assuming that the entries of the matrix M aremuch larger than the electroweak scale one can integrate out the heavy right-handedneutrino fields. One then obtains the standard electroweak theory supplemented bythe effective dimension five operator

Yν M−1Y Tν (εi j Hi l j )(εkm Hklm) (12.41)

(from tree diagrams with νcL exchange) which, after the Higgs field aquires a non-

zero vacuum expectation value, gives masses to the almost pure Majorana left-handed neutrinos. This see-saw mechanism (Gell-Mann, Ramond & Slansky 1979)can naturally explain the neutrino masses in the experimentally suggested rangeprovided the Majorana mass M is of the order of the Grand Unification scale.

We can now rewrite the matter couplings (12.14) to the gauge bosons in termsof the physical fermion fields:

Lmatter ⊃ − g√2(J−µ W+µ + J+µ W−µ)− g

cos θWJµ

Z Z0µ − eJµ

EM Aµ (12.42)

where (returning first to the Weyl spinor notation – rewrite (12.40) in terms of theleft-handed Weyl fields)

J−µ =∑

A

(u AσµdA + νAσµeA

)=∑AB

u′Aσµ(VCKM)ABd ′B +∑

A

ν ′Aσµe′A

≡∑AB

� ′u A

γµPL(VCKM)AB�′dB+∑

A

� ′νAγµPL�

′eA, (12.43)

and VCKM ≡ U†LDL is called the Cabibbo–Kobayashi–Maskawa (CKM) mixing

matrix. In (12.37) neutrinos are massless and, therefore, the neutrino fields canalways be rotated in such a way that there is no lepton mixing in (12.43). It isalso important that the matrices UR and DR drop out completely from the finallagrangian, because the right-handed fields only have diagonal couplings to the Bµ

gauge field. The neutral weak current and the electromagnetic current have thesame form in the mass eigenstate and in the electroweak basis. Note, in particular,that due to the doublet structure of all left-handed quark multiplets, the neutralweak current remains diagonal in flavour in the mass eigenstate basis. Finally, theYukawa couplings of the physical scalar h0 and of the unphysical pseudoscalarGoldstone boson G0 (present in the Rξ gauge) become diagonal in flavour in themass eigenstate basis. Indeed, they are determined by the same Yukawa coupling

12.3 Fermion masses and mixing 401

matrices which give the masses, and are diagonalized simultaneously with thelatter.† The coupling of the unphysical charged Goldstone boson G+ to physicalquarks involves the CKM matrix as for the charged weak currents. This completesthe discussion of the electroweak lagrangian in the Rξ gauge. The Feynman rulesfor the full lagrangian can be derived by the standard methods of Chapter 2, andare summarized in Appendix C.

We end this section with a few more remarks. The Cabibbo–Kobayashi–Maskawa quark mixing matrix is a 3 × 3 unitary matrix. In the general case of nquark doublets, the n×n unitary mixing matrix has 2n2−n2 = n2 real parameters,but not all of them are observable. Indeed, 2n − 1 phases can be removed byphase redefinitions of the physical quark mass eigenstates. Notice that the phasetransformations on the left- and right-handed quarks have to be the same to preservereal diagonal mass terms. Moreover, the matrix VCKM remains invariant when wechange all quarks by the same phase. So only 2n − 1 phases can be removed. Thenumber of real parameters is therefore n2−2n+1, of which 1

2 n(n−1) are angles ofn×n orthogonal matrix. Therefore, n2−2n+1− 1

2 n(n−1) = 12(n−1)(n−2) is the

number of phases. For n = 2 no physical phase in the CKM matrix is present. Forn = 3 there is one physical phase. In the world with three generations of quarksthe C P symmetry is explicitly broken if this phase is different from zero. The twomost often used parametrizations of VCKM read

VCKM = cθcβ sθcβ sβ exp (−iδ)

−sθcγ − cθsβsγ exp (iδ) cθcγ − sθsβsγ exp (iδ) cβsγ

sθsγ − cθsβcγ exp (iδ) −cθsγ − sθsβcγ exp (iδ) cβcγ

(12.44)

with the angles θ , β, γ and the phase δ fixed by experiment, and

VCKM ≈ 1 − 1

2λ2 λ Aλ3(ρ − iη)

−λ− iAλ5η 1 − 12λ

2 Aλ2

Aλ3(1 − ρ − iη) −Aλ2 − iAλ4η 1

(12.45)

The second one (Wolfenstein 1983) is only approximate (in this parametrizationVCKM is unitary only up to order O(λ4)), but is sufficient for the present level ofaccuracy of the measurements. We know from experiment that λ ≈ 0.23 (theCabibbo angle) and A ≈ 1, and from (12.45) we clearly see the order of magnitudeof the elements of the VCKM matrix.

† This is no longer true if the Higgs sector consists of several Higgs doublets, unless for some symmetry reasonsonly one doublet couples to the up-type quarks and one to the down-type. Thus, with several Higgs doubletsone generically has phenomenologically dangerous flavour changing neutral-current transitions mediated byHiggs boson exchange. This is another reason (apart from economy) to consider one-doublet theory.


12.4 Phenomenology of the tree level lagrangian

The electroweak theory discussed in this chapter correctly describes all knownphenomena induced by the electromagnetic and/or weak forces up to the energyscale O(100 GeV). For processes induced by the weak forces there are two regionsof energy scale which have been studied experimentally particularly thoroughly.These are: (a) very small momentum, q ≈ 0, processes (mainly weak decaysof leptons and hadrons); (b) the region of the Z 0 resonance at s1/2 ≈ 91 GeVstudied experimentally with very high precision in e+e− collisions at LEP1.† Weare going to concentrate on these two regions and to provide sample calculations inthe standard electroweak theory. This will illustrate the basic logic of such calcula-tions. One should also remember that weak decays of elementary particles played acrucial role in the development of the electroweak theory, with non-renormalizablefour-fermion lagrangian introduced on purely phenomenological grounds in 1933(Fermi theory).

The parameters of the electroweak gauge sector of the Standard Model la-grangian are the gauge couplings g and g′ and the vacuum expectation value v

of the Higgs field. Of course, they can be replaced by some combinations of them,and a convenient choice is, for example,

sin2 θW = g′2

g2 + g′2, α = g2 sin2 θW

4π, M2

Z =1

4(g2 + g′2)v2 (12.46)

Three independent measurements are thus necessary to fix these input parame-ters of the theory (beyond tree level they are renormalization-scale-dependent).Any further measurement is already a test of its predictions. At present thebest measured electroweak quantities are αEM, Gµ and MZ , where αEM =1/137.0359895(61) is the fine structure constant, Gµ = 1.16639(2)×10−5 GeV−2

is the so-called Fermi constant directly related to the muon decay width (see(12.50)) and MZ = 91.1867 ± 0.002 GeV is the physical Z0-boson mass(we distinguish it by the hat from the mass introduced as the parameter of thelagrangian; this distinction becomes important beyond the tree level). We shall usethese observables as the three measurements fixing the parameters of the lagrangianorder by order in perturbation theory, i.e. we need to calculate them to the requiredaccuracy in the electroweak theory parametrized by the set (12.46). It is clear fromSection 12.1 that at the tree level we get α = αEM, MZ = MZ . Moreover, as weshall see, sin2 θW is fixed in terms of Gµ, αEM and MZ .

The complete set of the electroweak parameters also includes fermion masses,CKM angles, the Higgs boson mass and the strong coupling constant. The fermionmasses are known with good (or very good) accuracy for leptons and less precisely

† At present (1999) the second phase of the LEP accelerator, the so-called LEP2 is operating, which will reachthe c.m. energy of 200 GeV, in a search for some departures from the Standard Model at those high energies.

12.4 Phenomenology of the tree level lagrangian 403

for quarks. One should also remember that the predictions for the electroweakprocesses in the hadronic sector generically have in addition uncertainties comingfrom our limited knowledge of the quark current matrix elements taken betweenthe hadronic states (i.e. from the lack of precise theory of hadrons as bound statesof quarks and gluons).

Effective four-fermion interactions

We proceed with the discussion of the tree level predictions. As advocated above,first we have to calculate the µ− decay widths, and then we shall be able to discussthe predictions of the theory.

All known elementary fermions, except for the top quark, are much lighterthan the W± and Z 0 bosons. Charge current processes with four external lightfermions, and with their momenta q � MW,Z , can be described in the tree levelapproximation by the effective four-fermion lagrangian

LCCeff (x) = −2

√2 GF J−µ (x)Jµ+(x) (12.47)

where

√2 GF = πα

s2c2 M2Z

(12.48)

We have abbreviated our notation, s ≡ sin θW, c ≡ cos θW, and used relation(12.25) (this relation is actually strictly valid for the parameters of the lagrangianin any order of perturbation theory in the MS scheme). The lagrangian (12.47)accounts for the amplitude that, for example, for the decay µ− → eνµνe,corresponds to the diagram in Fig. 12.1. The currents are the same as in (12.43).The V –A current is always built of a pair of fermions which form an SUL(2)doublet, and includes the appropriate CKM matrix element. It is important tostress that we introduce the effective four-fermion lagrangian as the four-point partof the integrand in the effective action (2.132), which is approximately local forthe considered range of phase space. The origin of this approximate locality isin the decoupling of the heavy particles (W±, Z0 and Higgs bosons and the topquark). This is a convenient way to organize certain calculations in the completeelectroweak theory, since we can discuss the whole class of processes at once. Theconcept of the effective low energy theory, in its full sense introduced in Chapter 7,will be discussed in Section 12.6.

At the tree level, the coefficient GF is universal for all four-fermion operatorspresent in the lagrangian (12.47). It can be determined in terms of the muon decaywidths. Calculating the muon decay (µ → eνµνe) width in the tree approximation


Fig. 12.1. Tree level diagram for the charged currents interactions in the Standard Model.

Fig. 12.2. Tree level diagram for muon decay in the Fermi theory.

we get

1/τµ = �µ = G2Fm5

µ/192π3 (12.49)

where τµ and �µ are the muon life time and its full decay width, respectively.Traditionally (but not quite consistently for a tree level calculation), the GF in

(12.48) is expressed in terms of the muon lifetime after correcting the theoreticalprediction for O(αEM) QED radiative effects. Historically, this calculation hasbeen performed in the Fermi theory, in a four-dimensional regularization schemeand with the on-shell renormalization (Berman 1958, Kinoshita & Sirlin 1959).Computing the full set of diagrams shown in Figs. 12.2 and 12.3 and integratingover the whole phase space, i.e. taking into account muon decays with one photonin the final state, one gets the following result:†

1/τµ = �µ =G2

µm5µ

192π3× f

(m2

e

m2µ

)[1 + αEM

2π

(25

4− π2

)](12.50)

where αEM is the fine structure constant, f (x) = 1 − 8x + 8x3 − x4 − 12x2 ln xand GF has been renamed Gµ. Actually, the standard attitude is that (12.50)defines a new fundamental constant, the so-called Fermi constant, which willbe denoted by Gµ. From this formula and the measured muon lifetime τµ =(2.19703 ± 0.00004) × 10−6 s the value Gµ = 1.16639(2) × 10−5 GeV−2 isextracted. Once GF is determined by its identification with Gµ, its comparison with

† This result is finite. In four-dimensional regularization schemes the counterterms shown in Fig. 12.3 cancelout. The absence of the IR and mass singularities follows from the Lee–Nauenberg–Kinoshita theorem(Chapter 5).

12.4 Phenomenology of the tree level lagrangian 405

Fig. 12.3. QED corrections to the muon decay in the Fermi theory. The crossed circlesdenote counterterms.

any other experimentally measured decay constant Gll ′ (defined by the analogueof (12.50) for beta decay of lepton l into lepton l ′) is a tree level test of theelectroweak theory (with the O(αEM) QED corrections included). As said earlier,at the tree level the parameters α and MZ of the lagrangian have to be identifiedwith the fine structure constant αEM and with the physical Z -boson mass MZ . Therelation (12.48), with GF = Gµ, can then be used to exchange the third parametersin2 θW for the accurately measured Gµ. With the three input parameters takenfrom experiment we can predict at the tree level the values of all other electroweakobservables.

For instance, for the W±-boson mass one gets (using again (12.25))

M (0)W = MZ

{12

[1 + (1 − 4A)1/2

]}1/2(12.51)

where A ≡ παEM/√

2 GµM2Z . Numerically, we get M (0)

W = 80.94 GeV (thesuperscript (0) means the tree level prediction for the physical W -mass). The recent(1997) combination of different measurements at LEP and Tevatron colliders givesMW = 80.39 ± 0.08. The discrepancy signals the need of including radiativecorrections (see next section).

The neutral-current transitions (like, for example, e−νµ → e−νµ) in the limitq ≈ 0, for vanishing fermion masses and in the tree approximation, can bedescribed by the effective four-fermion lagrangian

LNCeff = −ρ f f ′2

√2 Gµ Jµ

Z JZµ (12.52)


Fig. 12.4. Tree level diagram for neutral-current interactions in the Standard Model.

where the neutral current is

Jµ

Z =∑

f

� f γµPLT 3

f � f −∑

f

Q f sin2 θ efff (low)� f γ

µ� f

≡∑

f

� f γµ(g f

L PL + g fR PR)� f (12.53)

Calculating the Feynman amplitude corresponding to Fig. 12.4 and using (12.21)and (12.25) we see that at the tree level

sin2 θ efff (low) = sin2 θW (12.54)

and, moreover, we obtain the remarkable tree level prediction of the SM†

ρ f f ′ ≡ ρ = 1 (12.55)

This follows from the relation (12.25) (used in the derivation of (12.48)), which inturn is the consequence of using only Higgs doublets to break SUL(2) × UY(1)symmetry. In this case the Higgs sector has the so-called custodial SUV(2)symmetry (see Problem 12.3). These predictions can be compared with experi-mental determination of the parameters ρ (strictly speaking ρ f f ′) and sin2 θ eff

f (low)

defined by the lagrangian (12.52), interpreted as a low energy phenomenologicallagrangian. For instance, the parameter ρ measured in the leptonic sector of theneutral-current transitions is ρEXP = 1.006 ± 0.034 (Vilain et al. 1994).

Z 0 couplings

For the on-shell Z0 boson coupled to leptons and quarks we have (by analogy with(12.52); the differences are easy to understand)

LZ f f = −(ρ f√

2 Gµ)1/22MZ Jµ

Z Z0µ (12.56)

where, as before, Jµ

Z is given by (12.53) with

g fL = T 3

f − Q f sin2 θ efff , g f

R = −Q f sin2 θ efff (12.57)

† Note the factor of 2 in the denominator in the effective coupling 2√

2 Gµρ = g2/2M2Z cos2 θW obtained from

the tree level diagram. It follows from the identity of the two currents.

12.5 Beyond tree level 407

(one also often defines vector and axial couplings gV = gL + gR, gA = gL − gR).At the tree level one gets

sin2 θ efff = sin2 θ eff

f (low) = sin2 θW (12.58)

Also, then ρ f = ρ = 1.The decay width of Z0 into a fermion–antifermion pair is

�(Z0 → f f ) = Ncρ f√

2 GµM3Z

12π

(1 − 4m2

f

M2Z

)1/2

×[(g f 2

L + g f 2R )

(1 − m2

f

M2Z

)+ 6g f

L g fR

m2f

M2Z

](12.59)

where Nc = 1(3) for leptons (quarks). Thus, at the tree level we get �(Z0 →e+e−) = 84.82 MeV, which should be compared with the LEP measurement83.91±0.10 MeV. The experimental values for the parameters ρ f and sin2 θ eff

f canbe obtained by measuring the widths and various asymmetries in the Z 0 → f fdecays. The importance of higher order effects can be judged from the comparisonof the experimental world average sin2 θ eff

e = 0.23151±0.00022 with the tree levelprediction sin2 θ eff

f = sin2 θW = 12 [1−(1 − 4A)1/2] = 0.21215, where A is defined

below (12.51).

12.5 Beyond tree level

Renormalization and counterterms

The electroweak theory is a renormalizable field theory. A formal proof of this factcan be found in ’t Hooft (1971a, b). Crucial for renormalization is spontaneous(and not explicit) breaking of the gauge symmetry and also the matter content. Thelatter ensures the absence of chiral anomalies (Chapter 13). The interplay of quarksand leptons in obtaining the anomaly-free theory is a big puzzle of the StandardModel, which suggests the existence of a still deeper (more unified?) theory ofelementary interactions. In this book we shall be mainly concerned with practicalaspects of the renormalization programme to provide a basis for systematichigher order calculations. A vast literature exists on precision calculations andprecision tests of the electroweak theory which include all one-loop effects andalso, partially, higher order contributions. The complexity of the theory togetherwith arbitrariness in the choice of the ‘best’ renormalization scheme often make itdifficult to follow and to compare different approaches. Strictly speaking, part ofthe renormalization scheme is the choice of the input observables that are used tofix the parameters of the lagrangian. At present, after the experimental results ofLEP1, there is a consensus at least with regard to that latter point. The commonly


used input parameters are αEM, Gµ and the mass MZ of the Z0 boson. The remain-ing freedom in the renormalization scheme is the choice of the renormalizationconditions that fix the counterterms in the intermediate steps of the calculation.As in the rest of the book, we choose to work in the Minimal Subtraction scheme(actually in the MS scheme). Complete fixing of the renormalization scheme thenrequires only a choice of the renormalization scale µ. The Minimal Subtractionscheme is a convenient reference frame for comparison of different approaches.Any other renormalization scheme (for example, the often used on-shell OSscheme) is obtained by finite renormalization. Once we know the theory isrenormalizable, we are assured that calculation of any physical observable in termsof the input physical parameters (such as αEM, Gµ, MZ ) must give a finite answer.Therefore, in the one-loop calculations considered in this book, the most pragmaticattitude to the renormalization programme (useful in a theory as complex as theelectroweak theory) is to use lagrangian (12.5) as it stands simply without explicitlylisting the counterterms. Since we know the infinities cancel out, we can as wellsubtract them in the dimensionally regularized result, without keeping track of thenecessary counterterms. Thus, in one-loop calculations we can proceed as follows:

(1) Choose the set of the input observables in terms of which the others will be calculated.We use the set ai = (αEM, Gµ, MZ ).

(2) Compute all input observables ai , dimensionally regularized, in terms of the pa-rameters of lagrangian (12.5) interpreted as the renormalization scale-dependentrenormalized parameters in the MS (we shall use the set α(µ), s2(µ) and MZ (µ)

defined in (12.46)); keep only the terms which are finite in the limit ε → 0. We getα(1)EM = α

(0)EM + δαEM, M2(1)

Z = M2(0)Z + δM2

Z , G(1)µ = G(0)

µ + δGµ, in terms of the

same set of the renormalized lagrangian parameters. The quantities α(1)EM, G(1)

µ and

M2(1)Z are then identified with experimentally measured values αEM, Gµ and M2

Z , andthe equations can then be solved for α(µ), s2(µ) and MZ (µ). Of course, at the treelevel we have

α(0)EM = α, M (0)

Z = MZ , G(0)µ = πα√

2 s2(1 − s2)M2Z

(12.60)

and in this order one gets

c2 = 1 − s2 = 1

2

[1 +

(1 − 4παEM√

2GµM2Z

)1/2]≡ c2

0 (12.61)

(3) Compute in the one-loop approximation any other observable quantity of interest interms of the same set of the renormalized parameters of the lagrangian; keep only theterms which are finite in the limit ε → 0.

(4) Obtain finite predictions for the computed observables in terms of the chosen physicalinput observables αEM, Gµ and MZ by inserting into the result of step (3) theexpressions for the parameters α(µ), s2(µ) and M2

Z (µ) obtained in step (2).


The above prescriptions define the one-loop perturbative calculation up to someterms of the second order which may appear depending on the particular way ofrealizing this programme. One way of organizing such calculations is as follows(Barbieri 1993): calculate a renormalized matrix element (or any observablequantity)

M(1) =M(0)(α(µ), M2Z (µ), s2(µ))+ δM(α(µ), M2

Z (µ), s2(µ)) (12.62)

Using the relations M2Z ≡ M2(0)

Z = M2Z−δM2

Z , where MZ is the measured physicalmass, α(µ) ≡ α

(0)EM = αEM − δαEM and similarly replacing s2(µ) by the Fermi

constant, we can rewrite the last equation as follows:

M(1)(αEM, Gµ, MZ ) =M(0)(αEM, Gµ, MZ )+ δM(αEM, Gµ, MZ )

− ∂M(0)

∂αEMδαEM − ∂M(0)

∂Gµ

δGµ − ∂M(0)

∂M2Z

δM2Z (12.63)

For higher order calculations it is necessary to go beyond the ‘short cut’approach described above and to introduce the counterterms explicitly (see theend of Section 4.3). This is also helpful in getting a more profound flavour ofthe renormalization programme (for example, Ward identities) and is necessaryin studying the renormalization group equations for the parameters of the theory(Problem 12.4). The gauge invariance of the theory is easily taken care of ifwe introduce the renormalization constants at the level of the original lagrangian(12.5). We summarize here the renormalization constants of the complete StandardModel. For the gauge fields and couplings we follow the definitions introducedin Chapter 8 for QCD but we change the notation in the QCD sector for amore convenient one in the complete Standard Model framework (gs is the QCDcoupling constant, Ga

µ denote the gluon fields, the counterterms Z3 and Z1Y M arerenamed as ZG and ZS , respectively; the superscript B means ‘bare’):

GaBµ = Z1/2

G Gaµ

W aBµ = Z1/2

W W aµ

BBµ = Z1/2

B Bµ

H B = Z1/2H H

gBs = Z−3/2

G Zs gs

g′B = Z−3/2B Zg′g′

gB = Z−3/2W Zgg

(12.64)

The renormalization constants for the non-abelian couplings g and gs are definedas in Chapter 8. It is convenient to define the abelian Zg′ in the same way. Itis helpful at this point to recall the notation for QED in Chapter 5: there eB =


(Z1/Z2)Z−1/23 e ≡ Zee, where Z1, Z2, Z3 are the vertex, electron and photon self-

energy renormalization constants, respectively. The U (1) Ward identity ensuresZ1 = Z2 and Ze = Z−1/2

3 . If for QED we redefine as in (12.64), eB = Z−3/23 Zee,

we get Ze = Z3. In the present case the same argument applies separately to theleft- and right-handed chiral parts of the UY(1) lagrangian, and it follows that therelation Z B = Zg′ can be imposed.

For fermion fields we use the chiral notation of (12.2):(�lA

)BL = (Z1/2

l )A(�lA

)L(

�eA

)BR = (Z1/2

e )A(�eA

)R(

�qA

)BL = (Z1/2

q )AC(�qC

)L(

�u A

)BR = (Z1/2

u )AC(�uC

)R(

�dA

)BR = (Z1/2

d )AC(�dC

)R(

Y Al

)B = Z−1/2H (Z−1/2

e )A(Z AYl

Y Al )(Z−1/2

l )A(Y AC

u

)B = Z−1/2H (Z−1/2

u )AX (ZY uY )XY (Z−1/2q )Y C(

Y ACd

)B = Z−1/2H (Z−1/2

d )AX (ZY dY )XY (Z−1/2q )Y C

(12.65)

Since there is no flavour mixing in the lepton sector, the Yukawa matrices Y Al are

diagonal.† The quark wave-function renormalization constants are hermitean. TheYukawa coupling renormalization is determined by the vertex counterterms ZY sdefined by the matrix equations, for example, Y B

u �BR�

BL H B = ZYu Yu�R�L H , and

it is convenient to define the renormalization constant matrices δY s, for example,ZYu Yu = Yu + δYu . In Appendix C the vertex counterterms are then expressed interms of δY s.

Finally, for the Higgs boson mass parameter m2, its self-coupling λ and thevacuum expectation value v we write‡

vB = Z1/2H (v − δv)

λB = Z−2H (λ+ δλ)

(m2)B = Z−1H (m2 + δm2)

(12.66)

The additive notation is again useful, in particular since λ is additively renormal-ized (for example, by fermionic loops proportional to the Yukawa couplings).Replacing the unrenormalized fields and parameters in the original lagrangian(12.5) with the help of the above relations and splitting all Zs: Zi = 1 + δZi ,

† With massive neutrinos it is still convenient to work in the basis with diagonal matrix Y Ai and non-diagonal

effective neutrino mass operator m AB (εi j liA H j )(εi j l

iB H j ).

‡ In the Rξ gauge it is convenient but not necessary to introduce the counterterm δv for the Higgs vacuumexpectation value.


we can generate the necessary counterterms. (We do not explicitly write downthe renormalization constants for the ghost fields but they are included in theFeynman rules given in Appendix C.) We are already familiar with the fact thatthe counterterms which have been introduced at the level of the original lagrangian(12.5) are also sufficient to renormalize this theory after spontaneous breaking ofthe gauge symmetry. The Feynman rules derived from the lagrangian including thecounterterms, and in the four-component Dirac notation for fermions (appropriateafter spontaneous gauge symmetry breaking) are presented in Appendix C.

It is worth stressing once again that, going beyond the tree level, we must care-fully distinguish parameters of the lagrangian which depend on the renormalizationscale (and may also be gauge-dependent) and the physical quantities. For instance,MW and MZ must be distinguished from the µ-dependent lagrangian parametersMW and MZ (the same remark applies to the other masses too) which are givenby (12.24). We recall that in the MS scheme, in any order of perturbation theory,c2(µ) = M2

W (µ)/M2Z (µ). Likewise, the lagrangian parameter s2(µ) ≡ sin2 θW

defined by (12.18) should be distinguished from several quantities directly relatedto experimental measurements and often also called in the literature the Weinbergangles. Important examples are various sin2 θ eff introduced in Section 12.4 and theratio of the physical gauge boson masses 1 − M2

W/M2Z . To avoid any confusion,

in this book we shall never call the latter a sine of the Weinberg angle. Finally, werecall that, if the same physical quantity is calculated in terms of the same inputphysical observables to the same order in perturbation theory, the result does notdepend on the remaining freedom in the choice of the renormalization scheme. If,however, a given observable is calculated in terms of two different sets of inputobservables, finite order results can numerically depend on the choice of the inputparameters. Although this dependence is, in principle, of higher order, to estimatethe uncertainty one would have to perform full next order calculation.

In the following we discuss certain building blocks of the one-loop quantumcorrections in the electroweak theory, and then consider a sample of the mostrelevant physical applications. In particular, as we shall see, it will be convenientalso to recast one-loop corrections to the selected processes in the form ofcorrections to the effective lagrangians introduced in Section 12.4.

Corrections to gauge boson propagators

In general, radiative corrections to the processes described in Section 12.4 can beconveniently divided into two classes: universal corrections to the gauge bosonpropagators (called sometimes the ‘oblique’ corrections) which in most casesgive the dominant contribution, and process-dependent corrections consisting ofcorrections to the external fermion lines, corrections to the vertices and corrections


Fig. 12.5. Contributions to gauge boson self-energy.

generated by box diagrams. Process-dependent corrections are usually (but notalways) in the commonly used gauges, like, for example, Feynman (ξ = 1)or Landau (ξ = 0) gauge, much smaller than the ‘oblique’ corrections, andare therefore frequently neglected. One should, however, keep in mind, thatthe splitting into universal and process-dependent parts is gauge-dependent andto get physical, gauge-independent results one should include both types ofcorrection. Also, in general, the corrections to the gauge boson propagators are notseparately renormalization scheme-independent (in a given order) since they sharethe counterterms, for example, with the vertex corrections. Thus, in the calculationin the MS scheme, in which only 1/ε terms are subtracted, the dependence on therenormalization scale µ of the loop corrections to gauge boson propagators alonedoes not disappear. Moreover, in some schemes (like the MS scheme) inclusionof at least part of the process-dependent corrections may be necessary to avoidspurious singularities in the amplitudes in some kinematical limits (see below).

We begin with the process-independent corrections to the gauge boson propaga-tors. The full vector-boson self-energy amplitude, which consists in general of twoparts corresponding to diagrams shown in Fig. 12.5, will be denoted by†

i�µν

V1V2(p) = iPµν�T

V1V2(p2)+ iLµν�L

V1V2(p2) (12.67)

where Pµν and Lµν are transverse and longitudinal projection operators. Thetadpole diagrams must be included if one wants to verify gauge invariance ofthe mass counterterms. They cancel out, however, in physical amplitudes andwill be omitted in the following. Moreover, the longitudinal terms �L

V1V2of the

gauge boson self-energies do not contribute to the physical matrix elements. Thisis immediately seen for processes with external vector bosons (εµkµ = 0) andwith external fermions in the limit m2

f → 0 (conserved currents). In the generalcase these terms cancel (order by order in perturbation theory) Goldstone bosonself-energy amplitudes as a consequence of the Slavnov–Taylor identities.‡

† Note the sign convention which is different from the one used for fermion self-energies. This is convenientsince the free fermion and boson propagators have opposite signs and in both cases, after summing up one-particle reducible diagrams, the self-energy amplitudes at k2 = M2

V are corrections to the physical masses.‡ The gauge dependence of �Ts is cancelled by the vertex corrections. Note also that, for on-shell amplitudes,

the gauge-dependent parts of the tree level propagators cancel with the gauge-dependent Goldstone bosonpropagators. The net result is equivalent to using the propagators in the unitary gauge.


Fig. 12.6. Vector boson contribution to the vector boson self-energy.

As an illustration, we explicitly compute in the ’t Hooft–Feynman gauge thecontribution of the gauge boson loop to the gauge boson self-energies. The formu-lae for other contributions (from fermion, Higgs boson and mixed Higgs–vectorboson loops) are given in Appendix D. The diagram to be evaluated is shown inFig. 12.6. Using the trilinear vector boson couplings given in Appendix C we getthen for Fig. 12.6 (in n = 4 − ε dimensions, with the dimensionless couplingsg → gµε/2)

(ig1)(ig2)µε

∫dnk

(2π)n

−igσσ ′

(k + p)2 − M21

−igλλ′

k2 − M22

× [gµσ (k + 2p)λ + gµλ(k − p)σ − gσλ(2k + p)µ

]× [

gνσ ′(k + 2p)λ′ + gνλ′(k − p)σ

′ − gσ ′λ′(2k + p)ν]

where the couplings are, in principle, renormalization scale-dependent but to thisorder of calculation they can be replaced by their tree level values. With somealgebra (and remembering that gµνgνµ = n) we find

−ig1g2µε

∫dnk

(2π)n

i

[(k + p)2 − M21 ][k2 − M2

2 ]

[gµν(2k2 + 2kp + 5p2)

+ (4n − 6)kµkν + (n − 6)pµ pν + (2n − 3)(kµ pν + pµkν)]

�TV1V2

(p2) and �LV1V2

(p2) can now be expressed in terms of the functions a, b0, Aand B given by (D.1)–(D.11). For instance,

�TV1V2

(p2) = − g1g2

(4π)2

[10A(p2, M1, M2)+ (4p2 + M2

1 + M22 )b0(p2, M1, M2)

+ a(M1)+ a(M2)+ 2M21 + 2M2

2 − 23 p2

]With some labour the other contributions to the vector boson self-energies can be

obtained in a fairly straightforward manner, using our experience from the chapterson the scalar field theory and on QED. We recall in particular that, due to UEM(1)


gauge invariance, only the transverse part of the photon propagator is renormalizedand, moreover, the photon self-energy can be written as (see Section 5.2)

i�µνγγ (p) = i

(gµν − pµ pν

p2

)p2�γ γ (p2) (12.68)

i.e. �Lγ γ (p2) ≡ 0 and �T

γ γ (0) = 0. Using the results of Section 5.2 we can includethe fermion loop corrections to the photon propagator and, for instance, in the limitp2 = 0, which is useful for us one gets the following result:

�γ γ (0) = α

3π

(−23

4η + 1

2− 5

2ln

M2W

µ2+ 1

4ln

M2W

µ2+∑

f

Nc f Q2f ln

m2f

µ2

)(12.69)

Here Nc f and Q f are the colour factor (1 for leptons, 3 for quarks) and charge of thefermion, respectively; the two MW -dependent terms are the W± and the Goldstoneboson contributions, µ is the MS renormalization scale and the divergence η ≡2/(4− n)+ ln(4π)− γE may be removed by adding appropriate counterterms. Ofcourse, M2

W = c2 M2Z and all the parameters are renormalized at the scale µ, but to

this order of calculation they can be replaced by the physical parameters αEM, M2Z

and the appropriate function of Gµ.Another self-energy which requires a comment is the γ –Z0 mixing amplitude.

At one loop it is given by the same set of Feynman diagrams (Figs D.1 and D.2)as the corrections to the photon self-energy with appropriate changes in one of thevertices (see Appendix C). It will prove useful to decompose it as follows

�Tγ Z (p2) = �T

γ Z (0)+ p2�γ Z (p2) (12.70)

In the case of the photon self-energy, the contributions of the diagrams shown inFigs D.2(a)–(c) and D.1(a)–(c) plus D.2(d) separately vanish at p2 = 0. For theγ –Z 0 mixing amplitude diagram D.2(d) (with W+G− and W−G+ circulating inthe loop) does not cancel the contribution of D.1(a)–(c) at p2 = 0. Comparingwith the �T

γ γ case and isolating the combination of the two point functions whichdoes vanish at p2 = 0, one easily finds

�Tγ Z (0) = −2

α

4π

c

sM2

Z

(−η + ln

M2W

µ2

)(12.71)

It is important that �Tγ Z (0) does not contain a fermionic contribution.

Eight self-energies: �T(L)W W (p2), �

T(L)Z Z (p2), �

T(L)γ Z (p2), �T(L)

γ γ (p2) are madefinite by including counterterms shown in Appendix C, which depend on fiveindependent renormalization constants δZW, δZ B , δZg, δZH and δv. This is anon-trivial and practical test of the computation.


Fig. 12.7. ‘Oblique’ correction to muon decay and elastic neutrino scattering.

Fig. 12.8. Corrections to elastic neutrino scattering which are singular at p2 = 0.

In the limit p2 ≈ 0 and neglecting terms O(m2f /M2

W ), the one-loop correctionsto the gauge boson propagators, inserted into four-fermion Green’s functions(Fig. 12.7), can be summarized in the form of corrections to the effective four-fermion lagrangians:

LCCeff (oblique) =

(1 − �T

W W (0)

M2W

)× LCC

eff (Born) (12.72)

LNCeff (oblique) =

(1 − �T

Z Z (0)

M2Z

)× LNC

eff (Born) (12.73)

where, as it is clear from (12.47) and (12.48), for example,

LCCeff (Born) ≡ − 2πα

s2c2 M2Z

J−µ Jµ+.

It is important to remark that an additional contribution which should in principlebe included is the one shown in Fig. 12.8(a). Being singular at p2 = 0, it cannotbe incorporated as it stands into the neutral-current effective lagrangian. However,the singularity at p2 = 0 of the diagram shown in Fig. 12.8(a) cancels against thesimilar singularity contained in the vertex correction shown in Fig. 12.8(b). Thesum of these two graphs is finite at p2 = 0, numerically small and can be neglected.


The first interesting physical result which we can already get at this point is that,with one-loop accuracy and neglecting corrections other than the ones to the W±

and Z0 gauge boson propagators, we have

ρ f ≡ ρ f − 1 ≈ �TW W (0)

M2W

− �TZ Z (0)

M2Z

(12.74)

Of course, ρ f , as any other physical observable, is calculable (finite) in termsof αEM, MZ , Gµ and the Higgs boson mass, i.e. a systematic application of therenormalization programme gives a finite (and, therefore, renormalization scale-independent) result for ρ f in each order of perturbation theory. However, asmentioned earlier, as a generic situation in the case of some truncations in thisprogramme, the above combination of the self-energy amplitudes is not separatelyrenormalization-scale-independent. Still, the dominant contribution to (12.74) isfinite by itself. It comes from the third generation fermion loops. Indeed, we havealready noted that ρ f = 0 in the limit of exact custodial SUV(2) symmetry. Thissymmetry is explicitly broken by the g′ coupling, the Higgs sector and the fermionmass splittings within each generation. The latter contribution is finite by itselfsince it vanishes when the splitting vanishes. The largest violation of the custodialSUV(2) originates from the large top quark–bottom quark mass splitting. Thus, thedominant contribution to ρ is readily calculable.

With the help of the formulae for the fermionic contribution to the gauge bosonself-energies collected in Appendix D one finds

�T(t,b)W W (0) = Nc

4π

α

s2

[1

12(m2

t + m2b)−

1

6

m2t m2

b

m2t − m2

b

lnm2

t

m2b

− 1

3a(mt)− 1

3a(mb)− 1

6(m2

t + m2b)b0(0,mt ,mb)− 1

3m2

t −1

3m2

b

]

�T(t,t)Z Z (0) = Nc

4π

α

2s2c2

{(1 − 8

3s2 + 32

9s4

)[−2

3a(mt)− 2

3m2

t

− 1

3m2

t b0(0,mt ,mt)

]+(−8

3s2 + 32

9s4

)m2

t b0(0,mt ,mt)

}

�T(b,b)Z Z (0) = Nc

4π

α

2s2c2

{(1 − 4

3s2 + 8

9s4

)[−2

3a(mb)− 2

3m2

b

− 1

3m2

bb0(0,mb,mb)

]+(−4

3s2 + 8

9s4

)m2

bb0(0,mb,mb)

}

(12.75)


where Nc = 3 is the colour factor. It is easy to check that

�T(t,b)W W (0)

M2W

− �T(t,t)Z Z (0)

M2Z

− �T(b,b)Z Z (0)

M2Z

= Nc

4π

α

4s2c2 M2Z

(m2

t + m2b −

2m2t m2

b

m2t − m2

b

lnm2

t

m2b

)(12.76)

This result is indeed renormalization-scale-independent (as for a finite contribu-tion), and since mt � mb (experimentally mt = 175.6 ± 5.5 GeV whereasmb ≈ 5 GeV) it can be approximated as

ρ f ≈ NcαEM

16πs2c2

m2t

M2Z

(12.77)

Expressing the Weinberg angle in terms of Gµ (in a one-loop calculation the treelevel formula can be used), we obtain in our approximation the first one-loopprediction of the electroweak theory: ρ f = 0.0096.

So far we have discussed the self-energies of the vector bosons in a systematicperturbation theory. At each order the physical mass of the W± boson is givenin terms of the running mass M2

W (µ) by the requirement that the real part of thetwo-point non-local vertex of the effective action (also called 1PI two-point Green’sfunction), introduced in Section 2.6,

�TW W (k2) = k2 − M2

W −�TW W (k2) (12.78)

vanishes at p2 = M2W .† This condition applies to the transverse part of �W W . There

are, however, important physical situations such as the phenomena at energies inthe vicinity of the physical masses of unstable particles, for example, the e+e−

scattering close to MZ (in the resonance region), where a systematic perturbationtheory result is insufficient, and should be replaced by the Breit–Wigner form ofthe full propagator. Since the transverse part of the full W± propagator is givenby (2.152) (or, equivalently, can be obtained by summing all 1PI diagrams), wesee that it indeed has the Breit–Wigner form with the decay width of W± given byIm�T

W W (k2 = M2W ) (we ignore here the subtleties related to the definition of the

mass of unstable particles and to the gauge-independent definition of their width).The imaginary part of �T

W W (k2) can be obtained by applying the Landau–Cutkoskyrule to the Feynman integrals or by taking the imaginary part of the final result (seeAppendix D).

† We have not yet given all the calculations necessary for solving this equation with one-loop accuracy in termsof αEM, Gµ and M2

Z . For that, we need the one-loop relation between M2W (µ) and the above three measured

parameters, which will be discussed later in this section.


For the longitudinal part of the W± propagator the situation is more complicatedas it receives a contribution from the W±–Goldstone boson mixing. The two-point vertex function � is now a 2 × 2 matrix and the longitudinal part of the fullpropagator is obtained by inverting this matrix. Of course, the physical mass andthe width of W obtained from this procedure are the same (due to Ward identities)as those defined for the transverse part.

The same procedure is necessary for the Z 0 boson propagator because of theZ0–γ mixing. We begin with the 2 × 2 matrix for the inverse propagator

�TZγ (k

2) =(

k2 −�Tγ γ (k

2) −�Tγ Z (k

2)

−�Tγ Z (k

2) k2 − M2Z −�T

Z Z (k2)

)(12.79)

Using 2 × 2 matrix inversion we find for the transverse parts of the Z0–γpropagators

GTZ Z (k

2) = 1

k2 − M2Z −�T

Z Z (k2)− [�T

γ Z (k2)]2

k2 −�γγ (k2)

∼ 1

k2 − M2Z −�T

Z Z (k2)

GTγ γ (k

2) = 1

k2 −�Tγ γ (k

2)− [�Tγ Z (k

2)]2

k2 − M2Z −�T

Z Z (k2)

∼ 1

k2 −�Tγ γ (k

2)

GTγ Z =

�γ Z (k2)

[k2 −�γγ (k2)][k2 − M2Z −�T

Z Z (k2)] − [�T

γ Z (k2)]2

∼ �γ Z (k2)

k2(k2 − M2Z )

(12.80)

Fermion self-energies

We compute also the gauge boson and scalar contributions to the fermion self-energy. Denoting the amplitude corresponding to the general self-energy diagramsshown in Fig. 12.9 as

−i� ≡ −i (�L /pPL +�R /pPR +�m) ≡ −i(�V /p +�Aγ5 /p +�m) (12.81)

we have for Fig. 12.9(a):

−i� = (i)2µε

∫dnk

(2π)n

i

[(k + p)2 − m2]

−igµν

[k2 − M2]

×γ µ(c�V − c�

Aγ5) (/k + /p + m) γ ν(cV − cAγ

5) (12.82)

Performing the Dirac algebra and using the formulae (D.1) and (D.8) we obtain the


Fig. 12.9. General vector boson and scalar contributions to a fermion self-energy.

result

(4π)2� = n − 2

2p2

[a(M)− a(m)+ (p2 + m2 − M2)b0(p2,m, M)

]×/p

(|cV + cA|2 PL + |cV − cA|2 PR)

−m(|cV|2 − |cA|2) n b0(p2,m, M) (12.83)

Similarly, for a scalar contribution (Fig. 12.9(b)) we find

(4π)2� = 1

2p2

[a(M)− a(m)+ (p2 + m2 − M2)b0(p2,m, M)

]×/p

(cLc�

L PL + cRc�R PR

)+m

(cLc�

R PL + cRc�L PR

)b0(p2,m, M) (12.84)

For example, applying the general formula (12.83) to the W± and Z0 correctionsto the electron self-energy we get in the limit me � MW,Z :

�eL(0) = α

(4π)

[1

2s2

(1

2+ ln

M2W

µ2

)+ (1 − 2s2)2

4s2c2

(1

2+ ln

M2Z

µ2

)]�eR(0) = α

(4π)

s2

c2

(1

2+ ln

M2W

µ2

)(12.85)

All the parameters are, as usual, the running parameters.

Running α(µ) in the electroweak theory

Following our general strategy, we now have to calculate loop corrections to theinput observables to relate the renormalization scale-dependent lagrangian param-eters to the measured αEM, MZ and GF. These corrections enter into the calculationof any other (predicted) observable if accuracy beyond the tree level is needed. It isalso important to stress that the renormalization-scale-dependent coupling α(µ) inthe complete electroweak theory is different from the analogous coupling in QED,which is the low energy effective theory obtained after integrating out the weak


Fig. 12.10. Corrections to the photon–electron interaction. The external photon is realwith the four-momentum k ≈ 0.

sector. This point will be discussed in detail in Section 12.6. When necessary,we shall keep the distinct notation αSM(µ) and αEM(µ). Here we want to find therelation between αSM(µ) and the fine structure constant αEM = 1/137, and thesubscript ‘SM’ will be omitted.

The fine structure constant can be defined by the classical cross section forThomson scattering (see, for example, Bjorken & Drell 1964)

σ = 8π

3

α2EM

m2e

(12.86)

In any order of perturbation theory the lagrangian parameter α(µ) can be expressedin terms of αEM and M2

Z by calculating the Thomson scattering in the limit ofk → 0 (k is the photon four-momentum) in the complete electroweak theoryrenormalized at some scale µ. At the one-loop level one has to compute thediagrams shown in Fig. 12.10 (the black blobs denote one-loop diagrams) andone-loop W± and Z 0 contributions to the electron self-energies.† The diagramsimilar to Fig. 12.10(a) but with the photon replacing Z 0, at k = 0 and with theelectrons on-shell is exactly cancelled by the photon contribution to the externalline renormalization (as demonstrated in Chapter 5). It is convenient to summarizethe result of this calculation, performed in the MS scheme, in a form such thatit will then be easy to compare it with (12.86). By explicit calculation one findsthat the result obtained in the complete electroweak theory, in the limit k = 0, isreproduced by the following tree level lagrangian:

Lloweff = − 1

4(1 + δzγ )Fρκ Fρκ + (1 + δzL2 )�ei/∂PL�e + (1 + δzR

2 )�ei/∂PR�e

− e(µ)(1 + δzL1 )�e A/ PL�e − (δm + mδzL)�e PL�e

− e(µ)(1 + δzR1 )�e A/ PR�e − (δm + mδzR)�e PR�e (12.87)

† Box diagrams with a gauge boson (γ or Z0) attached to the external electron lines vanish in the limit k = 0.


All the fields and parameters are, of course, those of the complete theory. In theMS scheme and for k = 0 we get

δzγ = −�γ γ (0)

e(µ)δzL1 = −e(µ)(1 − 2s2)

2sc

�Tγ Z (0)

M2Z

− e3(µ)

(4π)2

[(1 − 2s2)2

4s2c2

(1

2+ ln

M2Z

µ2

)+ 3

2s2

(1

6+ ln

M2W

µ2

)](12.88)

e(µ)δzR1 = e(µ)

s

c

�Tγ Z (0)

M2Z

− e3(µ)

(4π)2

s2

c2

(1

2+ ln

M2Z

µ2

)(12.89)

where �γ γ (0) is given by (12.69) and �γ Z by (12.71). One-loop W± and Z 0

contributions to the electron self-energies have already been computed and aregiven by (12.85). They give

δzL = − e2(µ)

(4π)2

[1

2s2

(1

2+ ln

M2W

µ2

)+ (1 − 2s2)2

4s2c2

(1

2+ ln

M2Z

µ2

)]δzR = − e2(µ)

(4π)2

s2

c2

(1

2+ ln

M2Z

µ2

)(12.90)

Comparison with (12.86) is immediate if we rescale the fields in the lagrangian(12.87) to get the canonical form of the kinetic terms. The effective charge

e(µ)(1 + δzL

1 − δzL2 − 1

2δzγ) ≡ e(µ)

(1 + δzR

1 − δzR2 − 1

2δzγ)

(12.91)

can be identified† with αEM, and we get the one-loop relation

αEM = α(µ)

(1 + �γ γ (0)+ 2

s

c

�Tγ Z (0)

M2Z

)= α(µ)

[1 + α

4π

(2

3− 7 ln

M2W

µ2+ 4

3

∑f

Nc f Q2f ln

m2f

µ2

)](12.92)

We remind the reader once again that α(µ) ≡ e2(µ)/4π ≡ αSM(µ).One complication in using (12.92) is that �γ γ (0) contains light quark contri-

bution, and the result obtained for it in perturbation theory is not reliable. In thisregime strong interaction corrections become large and non-perturbative. To deal

† Two facts about (12.91) are worth noting: it is an equivalence relation (as we checked by explicit calculation)due to the unbroken UEM(1) symmetry and, moreover, δzL

1 − δzL2 = δzR

1 − δzR2 �= 0 because of the weak

contributions.


with this problem one writes

�γ γ (0) = �hadγ γ (0)+ �remainder

γ γ (0)

= −�hadγ γ (M2

Z )+ �hadγ γ (M2

Z )+ �remainderγ γ (0)

where �hadγ γ (M2

Z ) can already be computed in perturbation theory, and the finitequantity

�hadγ γ (M2

Z ) = �hadγ γ (M2

Z )− �hadγ γ (0) (12.93)

is the photon self-energy scalar function in the on-shell scheme. It can be computedvia the dispersion relation with one subtraction at p2 = 0 (since �(0) = 0). Thedispersion relation reads

�hadγ γ (s) =

1

πs∫ ∞

4m2π

ds ′Im �(s ′)s ′(s ′ − s)

(12.94)

and Im �(p2) can be expressed in terms of the measured ratio R(s) = σ(e+e− →hadrons)/σ (e+e− → µ+µ−) (where s is the centre of mass energy of the process):Im �(s) = 1

3αEM R(s). Evaluation of the integral gives the value �hadγ γ (M2

Z ) =0.0280 ± 0.0007 (Eidelman & Jegerlehner 1995).

Muon decay in the one-loop approximation

The most straightforward approach to using the muon decay width for fixing theparameters of the electroweak theory at the one-loop level is simply to expressthem in terms of the width itself. It is, however, much more elegant to considerthe constant Gµ defined by (12.50) as a fundamental constant, to write down theresult of the one-loop calculation in the electroweak theory in the same form andto express the lagrangian parameters in terms of the constant Gµ. This can beachieved if we succeed in splitting the full result into QED corrections to the treelevel effective lagrangian and the genuine one-loop electroweak corrections to thecoupling GF. Thus, in practice, we have to subtract from the full one-loop Green’sfunction the Green’s function obtained by calculating the diagrams in Figs. 12.2and 12.3. As we shall see, in the limit q ∼ 0 and neglecting fermion masses,the remaining piece can indeed be summarized in the form of the effective four-fermion lagrangian (in fact this is true in any order of perturbation theory). Thisis true for all charged current processes with four external fermions but, now, theeffective four-fermion lagrangian has to be written as a sum of flavour-dependent


pieces:

LCCeff = −

∑A,B,C,D

2√

2 G f(�u Aγ

µPL(VCKM)AB�dB

)(�dCγµPL(V

†CKM)C D�u D

)(12.95)

with similar terms (with obvious simplifications) for the leptonic and semi-leptonicprocesses. The subscript f of the couplings G denotes the dependence on all thefermion flavours of a given four-fermion operator. Due to the process-dependentcorrections, they are no longer universal and become fermion quantum numberdependent. Thus, for example, the lepton universality is no longer strictly trueand each decay: µ → eνeνµ, τ → µνµντ and τ → eνeντ has its own Fermiconstant. In the following we shall use Gµ (best measured experimentally) as theinput observable and the others will be the predictions of the theory at each orderof perturbation. These predictions can, for instance, be expressed as the values ofthe parameters κ f defined by the equations: G f = κ f Gµ.

For muon decay, we have to complete the calculation of one-loop correctionsto the relevant piece of the effective lagrangian (12.95) or, more precisely, tothe coefficient of the four-fermion operator

(�eγµPL�νe

)(�νµγ

µPL�µ

), which

describes muon decay. The subtle point of our subtraction procedure is that theresult (12.50) has been obtained with a four-dimensional regularization and inthe on-shell renormalization scheme, whereas now we work with the dimensionalregularization and in the MS scheme. One obvious consequence is that our finalresult (suppose it has the form (12.50)) will depend on the renormalization scaleµ: αEM in (12.50) will be replaced by α(µ). This difference is a higher order effectand can be neglected at one-loop accuracy. Another consequence is more delicate.

As mentioned earlier, with four-dimensional regularization prescriptions, thevertex counterterm and the wave-function counterterms in Fig. 12.3 cancel out.This is, in general, no longer true in dimensional regularization. By explicitcalculation one can check that the vertex correction to the lagrangian (12.47) hasthe structure

α

π

GF√2

(1

ε+ const.

) (�eγµγνγρ PL�νe

) (�νµγργνγµPL�µ

)≡ α

π

GF√2

(1

ε+ const.

)O1 = α

π

GF√2

(1

ε+ const.

)[4O2 + (O1 − 4O2)]

where O2 ≡ (�eγλPL�νe

) (�νµγ

λPL�µ

). The diagrams with the fermion self-

energy correction give

−4α

π

GF√2

(1

ε+ const.′

)O2


In four dimensions there is the identity†(�eγ

λγ κγ ρ PL�νe

) (�νµγ

ργ κγ λPL�µ

) = 4(�eγ

κ PL�νe

) (�νµγ

κ PL�µ

)(12.96)

and the two divergent contributions would cancel each other. The finite resultwould then be a finite correction to (12.47). However, the identity (12.96) is notvalid in 4− ε dimensions and instead for the sum of the vertex and the self-energycorrections we get an additional piece(

1

ε+ const.

)(O1 − 4O2) (12.97)

Our calculation requires, therefore, an additional counterterm in the form of theoperator (O1 − 4O2) and an additional renormalization condition. Thus, theoriginal MS scheme must be supplemented by this renormalization condition, andthe finite part in (12.97) depends on the choice of it. It is possible to chose thesubtraction scheme so that one subtracts all the 1/ε poles and, in addition, thesefinite pieces (Dugan & Grinstein 1991). There is then no need to calculate matrixelements of the (so-called evanescent) operator (O1 − 4O2). The finite result is thesame as if (12.96) was used in n dimensions. Another way of reaching the sameresult is to perform the one-loop QED calculations starting from the outset withthe tree level effective lagrangian written in the ‘charge retention’ form (12.98),i.e. after the Fierz rearrangement(

�eγλPL�νe

) (�νµγλPL�µ

) = (�eγ

λPL�µ

) (�νµγλPL�νe

)(12.98)

which again follows easily from (A.41).With the above background we can proceed with the subtraction procedure. The

finite parts of the last four graphs shown in Fig. 12.3 are equal to their obviousanalogue in the full amplitude and the latter should be simply omitted. The vertexgraph in Fig. 12.3 is in the SM replaced by the box diagram shown in Fig. 12.11.

† In the Weyl spinor notation it has the form(eσ λσκ σ ρνe

) (νµσ ρσκ σ λµ

) = 4(eσ κ νe

) (νµσ κµ

)and follows directly from the completeness relation for the σ matrices summarized in Appendix A (A.41). Inthe four-component notation it follows from the identity

γµγργν = γµgρν + γνgµρ − γρgµν − iεµρνσ γσ γ5

εαβγµεαβγ ν = −6gµν,

which imply

(γµγργν PL)i j (γνγ ργ µPL)kl = 4(γµPL)i j (γ

µPL)kl


Fig. 12.11. W –γ box in the Standard Model.

Hence, the result for that diagram should be subtracted from the one for the boxdiagram.

We calculate the contribution to the effective lagrangian from the box diagramshown in Fig. 12.11 in the limit of zero external momenta. The correspondingamplitude is finite and is reproduced by the following effective lagrangian:

LCCeff (γ W ) = i

e4

2s2

∫d4k

(2π)4

[�νµγ

λPL(/k + mµ)γρ�µ

] [�eγρ(/k + me)γλPL�e

][k2 − m2

µ][k2 − m2e][k2 − M2

W ]k2

In the limit of zero fermion masses and introducing the photon mass λγ toregularize the integral in the infrared one gets

LCCeff (γ W ) = e4

2s2

∫d4k

(2π)4

ikµkν

k4[k2 − M2W ][k2 − λ2

γ ]

× (�νµγ

λγµγρ PL�µ

) (�eγργνγλPL�e

)Replacing kµkν → (1/n)k2gµν → 1

4 k2gµν under the integral and using the identity(12.96) we get (see (D.15) and (D.18))

LCCeff (γ W ) = e4

2(4π)2s2F(0, MW , λγ )

(�νµγ

λPL�µ

) (�eγλPL�e

)= α

4πln

M2W

λ2γ

× LCCeff (Born) (12.99)

where LCCeff (Born) now denotes only that part of the tree level effective lagrangian

(12.47) which is relevant for the muon decay (we have retained only terms singularin the limit λγ → 0).

Computing in the same limit (again introducing the photon mass) the effectivelagrangian generated by the first diagram shown in Fig. 12.3 one gets

− α

4π

(1

2+ ln

λ2γ

µ2

)× LCC

eff (12.100)

where µ is the renormalization scale. As explained earlier, this result is obtainedeither by starting with the charge retention form of the four-fermion effective


Fig. 12.12. Remaining box diagrams for the muon decay in the Standard Model.

lagrangian, or by imposing the particular renormalization condition for the newcounterterm which is necessary in n dimensions if we work with the lagrangian(12.47). Subtracting (12.100) from (12.99) (and identifying to this order LCC

eff withLCC

eff (Born) in (12.100)) we finally get

LCCeff (γ W ) = α

4π

(1

2+ ln

M2W

µ2

)× LCC

eff (Born)

≡ Bγ W × LCCeff (Born) (12.101)

Note that the dependence on the IR cut-off has disappeared. The remaining boxdiagrams which have to be evaluated are shown in Fig. 12.12. The computation ofthe first two makes use of identity (12.96). The result is again the Born effectivelagrangian multiplied by the coefficients −(e2/16π2)(1/4s4) ln(M2

W/M2Z ) and

−(e2/16π2)[(1 − 2s2)2/4s4] ln(M2W/M2

Z ), respectively. The other two diagramsgive amplitudes which are equal to each other and, after using the identity

(�eγ

λγ κγ ρ PL�νe

) (�νµγ

λγ κγ ρ PL�µ

) = 16(�eγ

κ PL�νe

) (�νµγ

κ PL�µ

)(12.102)

each give −(e2/16π2)(2/s4)(1 − 2s2) ln(M2W/M2

Z ) as the coefficient of the effec-tive Born lagrangian. Together, these four box diagrams generate the effectivelagrangian

LCCeff (W Z boxes) = − α

4π

(1 − 5

s2+ 5

2s4

)ln

M2W

M2Z

× LCCeff (Born)

≡ BW Z × LCCeff (Born) (12.103)

Vertex corrections to be computed are shown in Fig. 12.13. In the limit ofvanishing external momenta and neglecting the fermion masses, when inserted inthe full diagram for the muon decay, these corrections give

LCCeff (vertex) = �× LCC

eff (Born) (12.104)


Fig. 12.13. Vertex corrections.

where � ≡ �(a) +�(b) +�(c) +�(d) and

�(a) = α

4π

1 − 2s2

4s2c2

(1

2+ ln

M2Z

µ2

)�(b) = − α

4π

3

2s2

(−5

6+ ln

M2Z

µ2− c2

s2ln

M2W

M2Z

)�(c) = − α

4π

3 − 6s2

2s2

(−5

6+ ln

M2Z

µ2− c2

s2ln

M2W

M2Z

)�(d) = − 3α

4π

(−5

6+ ln

M2Z

µ2+ ln

M2W

M2Z

)

(12.105)

Diagrams renormalizing external fermion lines are shown in Fig. 12.9. The W±

and Z0 gauge boson contributions to the muon and electron self-energies are givenby (12.85). For the neutrino self-energy, using the general formula (12.83), onegets

�νµ(νe)L(0) =α

4π

[1

2s2

(1

2+ ln

M2W

µ2

)+ 1

4s2c2

(1

2+ ln

M2Z

µ2

)](12.106)

where we have again neglected the charged lepton masses. In the calculation ofthe box, vertex and external wave-function renormalization corrections we alsoneglect the contributions of the Higgs and Goldstone bosons whose couplings areproportional to the masses of fermions to which they couple. Due to the presence ofthe projections PL in the µWνµ and eWνe vertices only �Ls contribute to the muondecay amplitude. We conclude that, up to terms O(m2

f /M2W ), the contribution of

the external line renormalization to the effective lagrangian for muon decay reads

LCCeff (ext lines) = 1

2

(�µL +�νµL +�eL +�νeL

)× LCCeff (Born) (12.107)


Collecting together all the corrections we get (note that corrections to theelectron and muon ‘sides’ are equal in the limit mµ,e → 0)

Bγ W + BW Z + 2 ×�+ 2 × 12

(�µL +�νµL

)= α

4πs2

(−4 ln

M2Z

µ2+ 6 + 7 − 12s2

2s2ln

M2W

M2Z

)(12.108)

Finally, adding the correction (12.72) to the W± boson propagator computedearlier we get

Gµ√2= πα(µ)

2s2c2 M2Z (µ)

[1 − �T

W W (0)

c2 M2Z (µ)

+ Bγ W + BW,Z + 2�+�µL +�νµL

](12.109)

After (12.92), this is the second equation which relates the measured quantities tothe running parameters. The third one is the one-loop correction to the physicalmass of the Z0 boson. From (12.79) and (12.80) we have

M2Z = M2

Z (µ)+�TZ Z (M2

Z ) (12.110)

The three equations can be solved for the parameters α(µ), s2(µ) and M2Z (µ) in

terms of αEM, M2Z and Gµ. This task is simplified if we notice that, to the order

we are working, we can replace the running parameters by the measured ones inall one-loop corrections. Eqs. (12.92) and (12.110) can immediately be inverted.Inserting the obtained α(µ) and M2

Z (µ) into (12.109) we get

Gµ√2= παEM

2s2c2 M2Z

(1 + ) (12.111)

where with one-loop precision

(µ) = −�TW W (0)

c20 M2

Z

+ �TZ Z (M2

Z )

M2Z

− 2s0

c0

�Tγ Z (0)

M2Z

− �γ γ (0)+ Bγ W + BW Z + 2�+�µL +�νµL (12.112)

and c20 and s2

0 are given by (12.61). Eq. (12.111) can be used to express s2(µ) interms of Gµ. We get

c2(µ) = 1 − s2(µ) = 1

2

[1 + (1 − 4A)1/2 − 2A

(1 − 4A)1/2

]= c2

0 −A

c20 − s2

0

(12.113)

and, as in (12.51), A ≡ παEM/√

2 GµM2Z = s2

0c20.

With these results we can predict any other electroweak observable. For


instance, we can obtain the one-loop prediction for the W± mass. Using theequation

M2W = M2

W (µ)+�TW W (M2

W ) = c2 M2Z +�T

W W (c20 M2

Z )− c20�

TZ Z (M2

Z ) (12.114)

and expressing c2(µ) in terms of Gµ, (12.113), one gets the final one-loop result:

M2W = c2

0 M2Z

[1 − �T

Z Z (M2Z )

M2Z

+ �TW W (c2

0 M2Z )

c20 M2

Z

− s20

c20 − s2

0

]

≡ c20 M2

Z

[1 − s2

0

c20 − s2

0

r

](12.115)

This result is, of course, renormalization-scale-independent, as it should be when aphysical observable is expressed in a given order of perturbation theory in terms ofother physical observables. It should be remembered that the important step wasthe use of the strictly one-loop relations (12.92) and (12.69) expressing αSM(µ)

in terms of αEM. Those relations contain logarithms, and there is no choice ofthe scale µ (which is arbitrary from the point of the final result (12.115)) whichcan make all of them simultaneously small. Thus, the result (12.115) suffers fromconsiderable inaccuracy introduced when expressing αSM(µ) in terms of αEM. Weshall return to this point in Section 12.6. We shall show there that it is possibleto calculate αSM(µ ≈ MZ ) in terms of αEM with all large logarithms summed upto all orders. Then using that αSM(MZ ) in (12.109) is an important improvement.Although it introduces only partial summation of higher order effects and, as such,µ dependence in (12.115), the numerical accuracy is improved since these arethe largest corrections to MW . In the last step of (12.115) we have defined thecorrection r , which is the common notation in the literature. From the previousequations we get

r = −�TW W (0)

c20 M2

Z

+ �TW W (c2

0 M2Z )

c20 M2

Z

− c20

s20

[�T

W W (c20 M2

Z )

c20 M2

Z

− �TZ Z (M2

Z )

M2Z

]

− �γ γ (0)− 2s0

c0

�Tγ Z (0)

M2Z

+ Bγ W + BW Z + 2�+�µL +�νµL (12.116)

Numerically, the most important contributions to r are contained in factors�γ γ (0) and [�T

W W (M2W )/M2

W ] − [�TZ Z (M2

Z )/M2Z ]. The second term in (12.116)

can be approximated as

r = · · · − c20

s20

ρ + · · · (12.117)


because(�T

W W (M2W )

M2W

− �TZ Z (M2

Z )

M2Z

)≈(�T

W W (0)

M2W

− �TZ Z (0)

M2Z

)≈ ρ (12.118)

With a bit more work, using the formulae for the b0 function, it is possible to showthat the leading mt dependence of r is:

r ≈ −c20

s20

3αEM

16πs20

m2t

c20 M2

Z

+ αEM

12πs20

lnmt

c0 MZ

(12.119)

For the dominant contribution of the Higgs boson to r one finds

r ≈ αEM

16πs20

11

3ln

M2h

c20 M2

Z

(12.120)

The terms ∼M2h/M2

W cancel out. This is an illustration of the important screeningof heavy Higgs boson effects (Veltman 1994).

In the limit of q ∼ 0 and vanishing fermion masses, loop corrections toflavour conserving neutral-current processes can be summarized in terms of theparameters ρ f Gµ and sin2 θ eff

f (low) already introduced in the lagrangian (12.52).They become flavour-dependent, i.e. different for different flavour pieces of thelagrangian (12.52) and relations (12.54) and (12.55) are modified. The custodialSUV(2) symmetry is broken by the Yukawa couplings and by the UY(1) current.

Corrections to the Z0 partial decay widths

A very precise test of the electroweak theory comes from the measurements of theZ0 boson properties at CERN and SLAC e+e− colliders.† It measures the partialdecay widths of the Z0 boson and various asymmetries of the decay products.All these observables are calculable from the effective lagrangian for the on-shellZ0 coupled to a pair of fermions, introduced in (12.56). Indeed, higher ordercorrections can also be summarized in terms of that effective lagrangian. Two-bodyZ0 decays can be described by a local effective lagrangian since they depend onkinematical invariants whose values are fixed in terms of the external particlemasses. Higher order effects can be parametrized by the effective parametersρ f Gµ and sin2 θ eff

f , which are different for different fermions and different from theanalogous parameters present in the low energy neutral-current lagrangian. Belowwe will compute radiative corrections to the effective Z0 f f lagrangian.

† For several years CERN LEP and linear SLAC electron–positron colliders operated at the centre of massenergy

√s = MZ producing millions of Z0s at rest and allowing for detailed study of its couplings to light

fermions.


Fig. 12.14. Diagrams for Z0 boson decay.

The relevant diagrams are shown in Fig. 12.14. Denoting the momentum of theZ0 boson by q and the momenta of the outgoing fermion and antifermion by p1 andp2, respectively, we have for these diagrams the following amplitudes (the factorT 3

f − Q f s2 in the couplings can be written as ± 12(1 − 2|Q f |s2) with the sign +

(−) for up (down) type fermions)

(a) = ∓ ie

2scu(p1)γ

λ[(1 − 2|Q f |s2)PL + (−2|Q f |s2)PR

]v(p2)ελ

(b) = ∓ ie

2scu(p1)γ

λ

[2sc|Q f |

�Tγ Z (M2

Z )

M2Z

]v(p2)ελ

(c) = ∓ ie

2scu(p1)γ

λ

[(1 − 2|Q f |s2)

1

2�T′

Z Z (M2Z )PL

+ (−2|Q f |s2)1

2�T′

Z Z (M2Z )PR

]v(p2)ελ

(d) = ∓ ie

2scu(p1)γ

λ [FL PL + FR PR] v(p2)ελ

(e)+ ( f ) = ∓ ie

2scu(p1)γ

λ[(1 − 2|Q f |s2)δZL PL + (−2|Q f |s2)δZR PR

]v(p2)ελ

The factors FL and FR denote one-loop vertex amplitudes (note that e/2sc hasbeen factored out), ελ(q) is the polarization vector of the Z 0 boson, �T′

Z Z (M2Z ) is

the derivative at q2 = M2Z of its transverse self-energy, and

δZL,R = �L,R(m2f )+ 2m2

f �′V (m

2f )+ 2m f �

′m(m

2f ) (12.121)

are the external fermion wave-function renormalization factors (see (12.81) and


compare with (5.42); each external wave-function renormalization factor is12δZL,R). Collecting all the factors together, expressing the MS parameters s, c ande through the measurable quantities (as we did in the case of r ) and replacingthe amplitude by the corresponding contribution to the effective action, the finalone-loop result can be written in the form (12.56) with

ρ f = 1 − +�T′Z Z (M2

Z )+ 2c0

s0

�Tγ Z (0)

M2Z

+ 2(FL − FR) (12.122)

Here

FL ≡ FL + (1 − 2|Q f |s20)δZL − 2

c0

s0

�Tγ Z (0)

M2Z

FR ≡ FR − 2|Q f |s20δZR

(12.123)

are the ‘renormalized’ (scale-independent) vertex corrections,† and is given by(12.112). Its presence originates from expressing with one-loop accuracy the factore(µ)/s(µ)c(µ) through measurable quantities. For the effective mixing angle wefind

sin2 θ efff = s2

[1 − c0

s0

�Tγ Z (M2

Z )

M2Z

− c0

s0

�Tγ Z (0)

M2Z

− FL +(

1 − 1

2|Q f |s20

)FR

](12.124)

where

s2 = s20

(1 + c2

0

c20 − s2

0

)(12.125)

The final result for the partial decay width is still subject to QED and (for f = q)QCD radiative corrections. With O(αEMαs) accuracy (including virtual and realemissions) we get

�(Z 0 → f f ) = �0(Z0 → f f )

(1 + 3αEM

4πQ2

f

)(1 − αs

π

), (12.126)

with �0 given by (12.59). For all fermions f except for the b quark‡ the vertexcorrections are small. The corrections to ρ f and sin2 θ eff

f are then dominated bythe factor ρ present in – see (12.74) and (12.118) – which contains a termproportional to m2

t /M2W (12.77), and which originates solely from expressing the

parameters s and c through s0 and c0. Since the effective coefficient of ρ ispositive, �(Z0 → f f ) increases with increasing top-quark mass.

† Indeed, in the popular on-shell renormalization scheme, where the counterterms also contain finite parts (fixedby some physical conditions), one finds precisely the combinations (12.123) for the renormalized vertexcorrections.

‡ For kinematical reasons Z0 cannot decay into a t t quark pair.


Fig. 12.15. mt -dependent vertex corrections to the Z0bb decay.

+ie√2s

mt

MW

(PR − mb

mtPL

)+i

e√2s

mt

MW

(PL − mb

mtPR

)

Fig. 12.16. Charged Goldstone boson couplings to the (t, b) quark pair.

For the bb pair in the final state there is an additional contribution of orderm2

t /M2W to the parameters ρb and sin2 θ eff

b . It comes from FL (the vertex correc-tions) and from the renormalization of the external b-quark line (diagrams (d)–( f )in Fig. 12.14). This is an interesting example of when the Appelquist–Carrazzonedecoupling theorem does not work. The reason is that the top-quark mass isgenerated by spontaneous breaking of the gauge symmetry and, therefore, largermass requires larger Yukawa coupling. We will compute the leading term. Therelevant vertex corrections are shown in Fig. 12.15. Inspection of the Feynmanrules in Fig. 12.16 and Appendix C reveals that only diagrams (e) and ( f ) containthe factor m2

t /M2W , coming from the Yukawa coupling G+tb of the Goldstone field

G+. Diagrams (c) and (d) are finite. The loop integral goes as 1/mt , compensatingfactors mt/MW coming from the G+tb vertices. Finally, diagrams (a) and (b) donot have any mt/MW factors in their vertices, and the logarithmically divergentloop integrals can at most give ln mt .


In the limit MW,Z � mt , i.e. for vanishing external momenta, and neglecting theb quark mass Fig. 12.15(e) gives†

ie

2scF (e)

L = e3

2s2

1 − 2s2

2sc

m2t

M2W

∫dnk

(2π)nPR

1

/k − mtPL

2kλ

(k2 − M2W )2

= −ie3

(4π)2

1 − 2s2

8s3c

m2t

M2W

[−η − 3

2+ ln

m2t

µ2+O

(1

mt

)]γ λPL

Similarly, for the Fig. 12.15( f ) we find

ie

2scF ( f )

L = e

2sc

e2

2s2

m2t

M2W

∫dnk

(2π)n

1

k2 − M2W

PR1

/k − mtγ λ

×[(

1 − 4

3s2

)PL − 4

3s2 PR

]1

/k − mtPL

= e3

4s3c

m2t

M2W

[−4

3s2 2 − n

n

∫dnk

(2π)n

k2

(k2 − M2W )(k2 − m2

t )2

+(

1 − 4

3s2

)∫dnk

(2π)n

m2t

(k2 − M2W )(k2 − m2

t )2

]γ λPL

= e3

4s3c

m2t

M2W

[−4

3s2 2 − n

n

∫dnk

(2π)n

1

(k2 − M2W )(k2 − m2

t )

+(

1 − 2

3s2

)∫dnk

(2π)n

m2t

(k2 − M2W )(k2 − m2

t )2

]γ λPL

= −ie

(4π)2

e2

8s3c

m2t

M2W

[4

3s2

(−η − 1

2+ ln

m2t

µ2

)+ 2

(1 − 2

3s2

)]γ λPL

Finally, the full contribution of order m2t /M2

W to the renormalization factor δZL ≈�L can be obtained by calculating the contribution of G± to the self-energy of theb quark. One obtains

�G±L (0) = 1

(4π)2

e2

4s2

m2t

M2W

[(−η − 3

2+ ln

m2t

Q2

)+O

(1

mt

)](12.127)

Collecting all contributions together (and replacing s, c by s0, c0) we get

2FL = 2

[FL +

(1 − 2

3s2

0

)�G±

L + · · ·]= − α

4πs20c2

0

m2t

M2Z

(12.128)

There is no similar contribution of order m2t /M2

W to FR. Comparing with ρ,(12.77), we find that for b quarks the new negative contribution (12.128) of order

† We also approximate the CKM matrix element Vtb by 1.

12.6 Effective low energy theory 435

m2t /M2

W to the parameter ρb overcompensates the positive universal contributionof the same order. As a result, �(Z 0 → bb) decreases as mt grows. The samecorrection (12.128) is also responsible for the slightly slower, compared to effectiveWeinberg angles for other fermions, decrease of sin2 θ eff

b with mt .

12.6 Effective low energy theory for electroweak processes

In Sections 12.4 and 12.5 we have presented a sample of systematic tree leveland one-loop calculations in the standard electroweak theory. We have used thelanguage of effective lagrangians, but that was merely as a local approximationto the effective action for the considered range of phase space. It is also veryinteresting to look at the previous results from another perspective, namely, tointroduce the effective low energy theory in its full sense defined in Section 7.7.In a renormalizable theory with several mass scales, the subsequent decouplingof heavy particles gives field theories which are valid in a smaller and smallerenergy range. In general, they are described by lagrangians which consist ofdimension four or less (renormalizable part) and higher dimension operators. Aneffective theory for light particles can be obtained by calculating the Green’sfunctions with only light field external legs at small momenta (compared toheavy masses) and with all heavy and mixed light–heavy particle loops. TheUV divergences arising in these graphs must be renormalized by appropriatecounterterms (for example, in the MS scheme). In the final step one redefinesthe light fields and parameters so that the whole dependence on the heavy fieldsis absorbed into those new, effective, parameters. They can be considered asgiven in terms of the parameters of the original theory (matching conditions) oras free, renormalizable parameters of the local effective theory. The main virtueof such a separation of mass scales is the possibility of easy resummation toall orders of large logarithms, ln(E/M), which one encounters when calculatinglow energy, E � M , processes in the full theory containing the large massscale M .

It is interesting to discuss this technique in the electroweak theory, with the muondecay amplitude as an example, and to compare it with the systematic perturbativecalculation performed in the previous section. Of course, summing up the termsO(α ln(MW/m f )) is less important than the terms O(αs ln(MW/m f )) (QCDcorrections) discussed in the next section, but it is interesting as an illustrationof the method.

Suppose we calculate at one-loop level the muon decay amplitude in thecomplete theory. This result has not been explicitly written down, but by inspectionof (12.109) and (12.50) one can convince oneself that it does not contain largelogarithms, ln(M2

W/m2µ), apart from those present in the running coupling α(µ).


This is pathological from the point of view of perturbative calculations in the fulltheory, which generically give such logarithms of two different mass scales, andthese are the logarithms that one would like to resum to all orders. Putting aside (fora moment) the pathology of our example, let us proceed with the construction ofthe effective theory, describing muon decay, by decoupling heavy particles: vectorand Goldstone bosons, Higgs bosons and the top quark. Clearly, at the tree level,such a procedure gives the effective theory described by the lagrangian

L = LQED − 2√

2 GF J+µ J−µ (12.129)

where GF is given by (12.49), and has now the interpretation of the Wilsoncoefficient.

The local, dimension six, operators present in lagrangian (12.129) are thelowest dimension operators describing weak transitions, obtained for four-fermionprocesses in the limit q → 0, i.e. after decoupling heavy particles. The Wilsoncoefficients of the effective four-fermion operators can be calculated in each orderof perturbation by comparing the leading term in the expansion in powers of(1/M2

W )n of the amplitudes calculated in the complete electroweak theory withthe amplitude generated by (12.129), with QED corrections included.† Thus, forinstance, to be able to identify the Wilson coefficient at one-loop accuracy, we haveto require equality of the full SM one-loop amplitude and the amplitude generatedby lagrangian (12.129), with one-loop QED corrections included. In fact, we knowalready the one-loop corrections to the Wilson coefficient GF: they are given by(12.109)! And here comes an important observation: the correction to the Wilsoncoefficient depends on the renormalization scale in the effective theory (rememberthat in the calculation of the previous section the scale µ is the renormalizationscale for QED corrections to the four-fermion operator) and involves a logarithmwhich can be large. It is minimized for µ ∼ MW .

In consequence, from the point of view of minimizing higher order corrections tothe Wilson coefficient GF(µ), the tree level effective lagrangian must be interpretedas the lagrangian at the decoupling scale µ ∼ MW , and (12.48) is the equation forthe Wilson coefficient GF(µ) at µ ∼ MW . The basic idea is then to apply the RGEin the effective theory to evolve GF(µ) down to the scale µ ∼ mµ, and to calculatethe muon-decay matrix element at that scale. In the present, pathological, exampleGF(µ) is renormalization-scale-independent at the one-loop level (a fact which isdirectly connected to the absence of logarithms of two separate scales in completeone-loop calculation), since the counterterm diagrams in Fig 12.3 cancel out. With

† A systematic expansion in inverse powers of M2W would also give higher dimension operators, for example,

m2f J−µ Jµ+, as well as terms with different chirality structure (V + A currents and scalar currents). Compared

to (12.129), they are suppressed by O(m2f /M2

W ), and for the light fermions are negligible compared to thecorrections O(αGµ) to the Wilson coefficients G f .


GF(MW ) = GF(mµ), we can finally calculate the muon-decay matrix element atthe tree level (to match the overall accuracy). Again, at the tree level that matrixelement does not depend on the renormalization scale but it should be understoodas taken at the scale µ ∼ mµ, to minimize higher order corrections to it. (Herewe encounter the same pathology: one-loop corrections to the matrix element, ofcourse, also do not contain any logarithms of the renormalization scale.) The maindifference of this approach compared to the tree level calculation in the full theoryis that α in (12.48), and in consequence in (12.51), is now α(MZ ). This is animportant improvement for calculating at the tree level electroweak observablesat the W± mass scale, for example, the W± mass, in terms of the low energyobservable Gµ.

In the next order of perturbation theory, we include one-loop corrections to GF,(12.109). Choosing µ ∼ MZ , we expect to minimize two-loop corrections. Next,for consistency, we should evolve GF(MW ) to GF(mµ) with two-loop RGE inthe effective theory. In this way we would resum all terms α(α ln(MW/m f ))

n)

of the full theory, if they had been present. In our pathological example GF isrenormalization-scale-independent at any order. One can see it by the followingargument. Suppose we calculate higher order QED corrections to the four-fermionoperator written in the charge retention form

(�eγ

λPL�µ

) (�νµγ

λPL�νe

), ob-

tained by the Fierz rearrangement, (12.98). This is a product of the neutrinocurrent (which does not interact with photons) and the current which, in thelimit of vanishing fermion masses, is conserved (and, therefore, does not undergorenormalization). Thus, all infinities generated by this four-fermion vertex can berenormalized by the field wave-function renormalization constants and the Wilsoncoefficient is renormalization-scale-independent (see Section 7.7). Finally wecalculate the muon decay matrix element by calculating finite parts of the one-loopQED corrections in the effective theory. Although those corrections do not containlogarithms of the renormalization scale, the latter should, in principle, be takenas µ ∼ mµ to minimize two-loop corrections to the matrix element (remember,however, that our example is pathological). Eventually we can express GF(MW ) interms of Gµ with the next-to-leading logarithm accuracy.

The perturbation theory based on lagrangian (12.129) would include higherorder contributions generated by perturbation in the four-fermion operator itself.Such contributions are effective dimension > 6 operators, with new dimensionfulcoupling constants ≤ O(G2

f ), and of course each new coupling requires newrenormalization constant (the theory is not renormalizable with a finite number ofcounterterms). The power of the effective theory approach resides in the truncationof this series, and in the present case we use LCC

eff only in the Born approximation,consistent with the fact that operators with dimension > 6 have already beenneglected in the construction of the effective theory.


QED as the effective low energy theory

It is now worthwhile to discuss in more detail the connection between the completeelectroweak theory and the renormalizable part of the effective theory defined bylagrangian (12.129), namely the QED of Chapter 5. Actually, even at the levelof pure QED we can discuss several steps of ‘effectiveness’: in the top-downapproach, starting with the decoupling of the weak sector and the even heavier topquark we obtain QED describing the interaction of photons, three lepton flavours(e, µ, τ ) and five quark flavours (u, d, c, s, b). This effective QED is valid forthe energy range 0 < E < MZ . However, it still contains hugely different massscales (fermion masses). Thus, the main virtue of effective theories (which is easyresummation of large logarithms) is achieved if this theory is used only in theenergy range mb < E < MZ . In the next step, we can integrate out subsequentquark and lepton flavours (in the order of their heaviness), obtaining each time anew renormalizable effective theory valid in a smaller and smaller energy range,and finally arriving at the QED of photons and electrons valid in the range 0 < E <

mµ. Each effective QED is described by its proper scale-dependent lagrangianparameters and, in particular, by its own coupling constant α

(ne,nd ,nu)EM (µ), where

ne, nd, nu are, respectively, the numbers of the charged leptons, down- and up-typequarks, present in the effective theory.

In the energy range of interest, the corresponding ‘minimal’ effective renor-malizable theory is not only sufficient but also best suited to describe physics inthat range (up to corrections O(q2/m2), where m2 is the next mass threshold).Eventually, our goal is to use each effective QED only in the energy rangemn < E < mn+1. Since the theory is renormalizable, its structure does notcontain any information about the energy range of its applicability. This is the usualbottom-up attitude in which we study physics in a certain energy range withouteven knowing the more complete theory. In the present case we are fortunate toknow both the complete theory and the effective theory, and we are able to prove bythe decoupling procedure that the usual bottom-up attitude is correct (Appelquist& Carrazzone 1975).

The first point to stress is that the renormalized (in the M S scheme) couplingconstant for each effective QED can be fixed in terms of αEM. In the one-loopapproximation we get:

αEM = α(ne,nd ,nu)EM (µ)

(1 + α

3π

∑ne,nd ,nu

Nc f Q2f ln

m2f

µ2

)+O

(m2

n

m2(n+1)

)(12.130)

These equations are obtained by calculating in each effective QED the k → 0 limitof the Thomson scattering (i.e. by repeating in these simpler cases the procedureused for calculating αEM in the complete electroweak theory). The result is


given by the corrections to the photon propagator. Of course, for fixed ne, nd, nu

the renormalized coupling is defined only in the range 0 < µ < mh , wheremh is the mass of the lightest decoupled fermion, and the µ scale in (12.130)can take any value in this range. We also remark that (12.130) is valid up tonon-perturbative corrections to the photon propagator discussed in Section 12.5.Eq. (12.130) reflects quite clearly the problem of large logarithm resummation.When we calculate the low energy observable, αEM, in an effective QED withseveral flavours, (12.130) contains large logarithms for any value of µ, and inthis theory improved accuracy can be achieved only by calculating higher ordercorrections to this equation.

However, we can proceed in a different way. It is clear from (12.130) that

αEM = α(1,0,0)EM (µ ≈ me)+O(α2), (12.131)

the result already known from Chapters 5 and 6. The accuracy of the determinationof α

(1,0,0)EM (µ2 ≈ m2

e) in terms of αEM is O(α2) since no large logarithm appearsin this case in the one-loop (12.130). (In fact, they are also absent in higherorder corrections to this relation used in the effective QED of only photons andelectrons.)

Once α(1,0,0)EM (µ) is fixed by (12.131) at some µ0 ≈ me, its µ dependence in the

MS is determined by the RGE. Summing up leading logarithms, i.e. in the one-loopapproximation for the β-functions, we have

µd

dµα

(1,0,0)EM (µ) = 1

2πb1(1, 0, 0)(α(1,0,0)

EM )2 (12.132)

where (in general)

b1(ne, nd, nu) = 43(ne + 1

3 nd + 43 nu) (12.133)

is the first coefficient of the β-function β(ne, nd, nu) with ne, nd, nu ‘active’fermions. For any µ in the range of the validity of the effective theory, 0 < µ <

mµ, the solution reads:

1

α(1,0,0)EM (µ)

= 1

α(1,0,0)EM (µ0)

− 1

4πb1(1, 0, 0) ln

µ2

µ20

(12.134)

Thus, we obtain α(1,0,0)EM at the scale µ = mµ, with all terms [α ln(mµ/me)]n

summed up. We can then switch to the next effective QED parametrized by therunning α

(2,0,0)EM (µ) (with β(2, 0, 0)), up to the next threshold etc., until we reach

α(3,3,2)EM (µ).In order to accomplish this programme, we have to know the relations between

the couplings of two different effective QEDs in the range of µ where both arevalid, for example, between α

(1,0,0)EM (µ) and α

(2,0,0)EM (µ) for µ ≈ m2

µ (for the sake of


our examples we assume that all quarks are heavier than leptons). One way to getthese relations is to calculate the same physical quantity in both theories. Taking itas αEM, with one-loop accuracy we get from (12.130):

α(1,0,0)EM (µ) = α

(2,0,0)EM (µ)

(1 + α

3πln

m2µ

µ2

)+O

(m2

e

m2µ

), (12.135)

for µ ≈ mµ. These are the so-called matching conditions. We see that in thissimple example, for µ ≈ mµ, the two couplings are equal up to correctionsO(α2). Such corrections can be neglected as long as we work with one-loop RGE.†(Another way to get (12.135) is to begin with two-lepton theory and to follow thedecoupling procedure. Then, α

1,0,0EM (µ) is the effective coupling constant of the

‘effective’ one-lepton theory obtained from the full two-lepton theory, and (12.135)illustrates the general feature of decoupling in the MS scheme: the dependence onthe heavy states is absorbed into effective parameters of the lower energy theory,which are free, renormalizable parameters from the point of view of the lattertheory.) It is very easy to see that our procedure finally gives

1

α(3,3,2)EM (µ)

= 1

α(1,0,0)EM (m2

e)+ 1

3π

∑f �=t

Nc f Q2f ln

m2f

µ2(12.136)

i.e. the leading logarithms are summed up to all orders (we remember thatα

(1,0,0)EM (m2

e) = αEM).The procedure which we have outlined here is important in the following

context. Suppose that, as in Section 12.5, we want to fix αSM(µ) in terms of αEM.The one-loop approximation, (12.92), contains large logarithms (there is no scaleµ for which all of them can be small). It is, therefore, very desirable to go beyondthe accuracy of (12.92) and to express αSM(µ) in terms of αEM with all the largelogarithms summed up to all orders. This is indeed achieved by using our effectivetheory approach.

The final step is to use the matching condition

αSM(µ) = α(3,3,2)EM (µ)

[1 − α

4π

(2

3− 7 ln

M2W

µ2+ 16

9ln

m2t

µ2

)]= α

(3,3,2)EM (µ)+O(α2), (12.137)

for µ ≈ MZ . Since at µ ≈ MZ the logarithms in (12.137) are small we can rewrite

† Of course, for µ = mµ we get α(1,0,0)EM = α

(2,0,0)EM , but this is not a necessary choice of µ. Eq. (12.135) can

be used for any µ such that (α/3π) ln(m2µ/µ2) � 1.

12.7 Flavour changing neutral-current processes 441

our result in the compact form:

1

αSM(MZ )= 1

αEM+ 1

(4π)

(2

3− 7 ln

M2W

M2Z

+ 4

3

∑f

Nc f Q2f ln

m2f

M2Z

)(12.138)

where the sum extends to all fermions including the top quark. This is the desiredimprovement over the one-loop formula (12.92).

In some very interesting physical questions, for example, in theories of grandunification, αSM(MZ ) is used as input parameter in the renormalization groupevolution up to scales µ ≈ 1016 GeV; see for example (Chankowski, Płuciennik& Pokorski 1995). Then, ln(µ/MZ ) � ln(m f /MZ ) and the accuracy of (12.138)requires for consistency the two-loop renormalization group evolution in that largeµ range.

Our final comment is that (12.137) illustrates two generic points. One is thatoften several mass scales must be decoupled at the same time. If both logarithmsare small enough the whole procedure is self-consistent. Another general fact isthe presence of some constant factors in the one-loop matching conditions. In suchcases the two-loop evolution of coupling constants is discontinuous. If one wishes,one can avoid the constant ( 2

3) in (12.137) (the so-called finite threshold correction)by using Dimensional Reduction (DR) scheme instead of the MS.

12.7 Flavour changing neutral-current processes

We have learned already in Section 12.3 that the structure of the electroweak theoryis such that it ensures the absence of the tree level flavour changing neutral currents.This follows from the SUL(2) × UY(1) gauge invariance, renormalizability andthe particle content of the theory. Both neutral gauge boson and Higgs bosoncouplings are diagonal in the flavour mass eigenstate basis. Moreover, the leptonnumber conservation (separately for each lepton flavour) is exact. It reflects theUe(1)×Uµ(1)×Uτ (1) global symmetry of the lagrangian (12.5). For quarks, onlythe global baryon number is conserved, while the quark mixing explicitly breaksthe quark flavour U (1)s. Thus, the flavour changing neutral-current processesinvolving quarks are generated in higher orders in the electroweak interactions.Since they are strongly suppressed in Nature, it is interesting to discuss thepredictions for them in the electroweak theory. Such processes are, of course,calculable without any new counterterms.

At the quark level, the generic examples of flavour changing neutral-currenttransitions are the reactions ds → ds ( S = 2), bd → bd ( B = 2) and s →dγ ( S = 1), b → sγ ( B = 1), with a real or virtual photon. Such transitionsare responsible for physical processes like K0–K0 and B0–B0 mixing (and, inparticular, for C P violation in the K0–K0 system), for radiative flavour changing


Fig. 12.17. (a) Leading Standard Model contribution to K0–K0 mixing; (b) penguindiagram contribution to a S = 1 process.

decays of strange and bottom mesons (like, for example, the celebrated b → sγdecay) and for decays like K → πe+e− or B → K�e+e−. Similar processes occurin the up-quark sector, too, but are less interesting phenomenologically.

Predictions for physical processes are obtained by convoluting the quark off-shell amplitudes with the hadron wave-functions or (in practice – as we shall see)by calculating the hadronic matrix elements of operators present in the effectivelagrangian. For the time being, we remain at the quark level and estimatequalitatively the expected order of magnitude for the S(B) = 2 and S(B) = 1amplitudes.

For the S(B) = 2 transitions, for example, for the sd → sd transition, we havethe contribution from the diagrams shown in Fig. 12.17(a) (G± denote the chargedunphysical Goldstone bosons). The two sets of diagrams enter with a relative minussign, as can be checked by general techniques developed in Chapter 2 for obtainingthe Feynman rules from a lagrangian, due to Fermi statistics of the quark fields. Weshall be interested in the off-shell 1PI Green’s functions which build the effectiveaction, (2.132). The corresponding contribution to the effective action is finiteand has dimension [mass]−2. In the limit of small (compared to the gauge bosonmass) external momenta which interests us, it is approximately local and can beinterpreted as an effective lagrangian. For the sum of the two diagrams 12.17(a)with the W+W− exchange, in the ’t Hooft–Feynman gauge, we have

Leff = 1

2

(e√2s

)4 ∑i, j=u,c,t

V �is Vid V �

js Vjd

×∫

d4q

(2π)4

[�sγµPL(/q + mqi )γν PL�d][�sγν PL(/q + mq j )γ

µPL�d]

(q2 − m2qi)(q2 − m2

q j)(q2 − M2

W )2

(12.139)

The colour indices are contracted within square brackets. Suppose we want to


recover the contribution of the effective vertex (12.139) to the Green’s function〈0|T (

�s�d�s�d) |0〉. We insert the effective lagrangian (12.139) into (2.202).

There are four possible ways of contracting the fields in Leff (12.139) withthe external fields that contribute to the 1PI four point Green’s function. Twocontractions give the two sets of Fig. 12.17(a), with the relative minus sign. Theother two contractions give the same result. Therefore, to recover the result of thecalculation in the original theory, we need an extra combinatorial factor of 1

2 infront of (12.139).

The contribution from the top quark exchange in (12.139) is strongly suppressedby its very small mixing with the first two generations of quarks. (As we shallsee, this suppression is just the right one for accounting for the measurements).As far as the u and c exchange is concerned, we can take the limit mqi � MW ,and make use of unitarity of the Cabibbo–Kobayashi–Maskawa (CKM) matrix.On dimensional grounds, we then get the following estimate for the sd → sdtransition amplitude with double W -boson, u- and/or c-quark exchange:

Auc ∼(

e√2s

)4 1

M2W

∑i, j=u,c

V �is Vid V �

js Vjd

[1 +O

(m2

qi

M2W

,m2

q j

M2W

)]

∼ αGF

{(V �

ts Vtd)2 +O

( ∑i, j=u,c

V �is Vid V �

js Vjd

m2q

M2W

)}(12.140)

In the last line, unitarity of the CKM matrix has been used:∑

i, j=u,c V �is Vid =

−V �ts Vtd . Consequently, the leading αGF term is suppressed by the same very small

CKM angles as the double top quark exchange contribution. The remaining terms,which are proportional to larger CKM angles, are in turn suppressed by light quarkmasses, i.e. they are O

(αGFm2

q/M2W

).† The same refers to contributions from G±

exchange, because couplings of G± to light quarks are suppressed by mq/MW .Such a mechanism of suppression of the flavour changing neutral-current

amplitudes is known as the Glashow–Iliopoulos–Maiani (GIM) mechanism. Wesee that the strong suppression of the flavour changing neutral-current transitionsis indeed a prediction of the Standard Model. However, this prediction follows notonly from the structure of the theory but also depends on the empirical patternof the quark masses and mixing angles. Thus, from the point of view of theStandard Model, the successful predictions for the flavour changing neutral-currentprocesses should be considered as fairly accidental and, in fact, it is probably the(unknown) physics beyond the Standard Model which is responsible for it (namely,the physics which determines the quark Yukawa couplings).

† The dimensionless coefficient of the term 1/M2W could also include the term ln(mq/MW ). For the box

diagram there is no logarithmic term. We shall return to this point shortly.


It is also interesting to comment on the S(B) = 1 transitions at the qualitativelevel. At one-loop, they receive contributions from box diagrams and also from theso-called penguin diagrams like, for example, the one shown in Fig. 12.17(b). Thecorresponding amplitude behaves as

Auc ∼ αGF

∑i, j=u,c

V �id Vis ln

m2qi

M2W

+O(V �td Vts) = αGFV �

ud Vus lnm2

u

m2c

+O(V �td Vts).

(12.141)This time, the dimensionless coefficient of the αGF term† contains logarithms

of light quark masses. We often say that the GIM mechanism is power-like in thecase of box diagrams, but only logarithmic in the case of certain penguin diagrams.(In practice, since the masses of the up and charm quarks are quite different, noadditional suppression except for the usual αGF factor occurs in this case, unlikethe previously considered box diagrams.)

This qualitative difference between penguin and box diagrams can be seen eitherby explicit calculation or by the following heuristic argument. The MW dependenceof the amplitude can be traced back to the behaviour of the effective field theorydiagrams which are obtained by contracting the heavy internal lines in the boxand penguin diagrams, respectively. In the first case we get either a tree level(UV-finite) diagram or a one-loop diagram with two four-quark vertices. Ondimensional grounds, the latter diagram is proportional to 1/M4

W , which meansthat the power-like GIM mechanism occurs. In the penguin case, we obtain aUV-divergent one-loop diagram with a single four-quark interaction. This signalsthe possibility of only logarithmic GIM mechanism.

In processes where the GIM mechanism is only logarithmic, perturbativecalculations can be trustworthy only if all the quark masses under logarithmsare larger than the QCD confinement scale. Thus (12.141) could be used in aquantitative calculation only if the up-quark mass was larger than ∼1 GeV. Sincethis is not the case in Nature, (12.141) can be considered only as an indicationthat non-perturbative strong interactions play an important role in the quark-levels → de+e− transition amplitude.

Since the flavour changing neutral-current transitions involve strongly inter-acting quarks, one of the most interesting field theoretical aspects are the QCDcorrections to them. They are often important phenomenologically, and theircalculation strongly relies on the effective theory approach. In the final subsectionof this chapter we describe these techniques using the concrete example of C Pviolation in the neutral kaon system.

† The leading term behaves like 1/M2W . The 1/k2 from the photon propagator is cancelled by the k2/M2

Wfactor coming from the vertex function. The logarithmic part of the vertex function is independent of quarkmasses and vanishes due to unitarity of the CKM matrix.


QCD corrections to C P violation in the neutral kaon system

We are interested in C P violation in the kaon–antikaon transition amplitude (forthe general discussion of C P violation in the kaon system consult Section 1.5).At the quark level, this amplitude is generated by the sd → sd amplitudewhich receives the leading-order contribution from the box diagrams presentedin Fig. 12.17(a). In the following, we shall first discuss the calculation of thesediagrams and QCD corrections to them. Next, we shall turn to the explicit form oftheir relation to the C P violation parameter εK.

The complete result for the Leff given by (12.139) is easy to obtain. Usingthe formulae from Appendix D and the identity (12.96), it can be written in thefollowing form:

Leff(W W ) = −G2F M2

Wλiλ j

4π2A(xi , x j )

(�sγµPL�d

) (�sγ

µPL�d)

(12.142)

with

A(xi , x j ) = 1

xi − x j

[x2

i ln xi

(1 − xi )2− (i → j)

]+ 1

(1 − xi )(1 − x j ). (12.143)

To this order, GF is given by (12.48) and xi = m2qi/M2

W . Summation over indicesi, j = u, c, t is understood in (12.142). The products of the CKM matrix elementshave been denoted shortly by λi = V ∗

is Vid and colour indices are contracted withinbrackets.

The remaining box diagrams (remember that (12.139) accounts only for the W –W exchange diagrams) with either two unphysical Goldstone scalars in the loop orone such scalar and one W±-boson are calculated in an analogous way. The sum ofthese diagrams is reproduced by an effective lagrangian which has the same formas Leff(W W ) with A(xi , x j ) replaced by

14 xi x j [A(xi , x j )− 8B(xi , x j )] (12.144)

where

B(xi , x j ) = 1

xi − x j

[xi ln xi

(1 − xi )2− (i ↔ j)

]+ 1

(1 − xi )(1 − x j )(12.145)

Thus, up to O(external momenta/MW ), the diagrams shown in Fig. 12.17(a) canbe reproduced by the following effective four-quark interaction lagrangian

Leff(box) = −G2F M2

W

4π2

∑i, j=u,c,t

λiλ j S(xi , x j )(�sγµPL�d

) (�sγ

µPL�d)

(12.146)where

S(xi , x j ) =(1 + 1

4 xi x j)

A(xi , x j )− 2xi x j B(xi , x j ) (12.147)


Fig. 12.18. Sample QCD correction to K0–K0 mixing.

The sd → sd amplitude presented in Fig. 12.17(a) is subject to QCDcorrections. They are described by diagrams obtainable from those shown inFig. 12.17(a) by connecting the quark lines by an arbitrary number of gluon lines,for example, like that shown in Fig. 12.18.

An important property of these corrections is that their ordinary perturbative ex-pansion is actually an expansion in powers of αs ln(M2

W/Q2low) and αs ln(m2

t /Q2low)

rather then in powers of αs itself. Here, Q2low stands for all possible Lorentz

invariants made out of momenta and masses of external particles in the consideredamplitude, i.e. it is at most of order of the neutral kaon mass mK � 0.5 GeVsquared. Since MW � 80 GeV and mt � 175 GeV, the logarithms arelarge, and the ordinary expansion in αs breaks down. We need to resum theselarge logarithms from all orders of the perturbation series. It is here where themain virtue of using the effective theory is, since we can then easily apply the(perturbative) renormalization group methods for resummation of large logarithms.This procedure will be described in some detail below.

The kaon–antikaon transition amplitude we look for is not perturbatively cal-culable even when large QCD logarithms are resummed. In order to find it, oneneeds to use non-perturbative methods like, for instance, lattice calculations. Onecould then ask why we intend to perform the perturbative logarithm resummationinstead of applying non-perturbative methods immediately. The answer to thisquestion is that the accessible non-perturbative methods are not suited for calcu-lating quantities which receive significant contributions from rapidly varying fieldconfigurations. In other words, gluons and quarks with momenta close to MW ormt (i.e. very much larger than �QCD) cannot be accounted for by non-perturbativemethods in the case when external states are light hadrons at rest. As we shall see,the purely perturbative resummation of large logarithms in the sd → sd amplitudewill automatically provide us with quantities suitable for applying non-perturbativemethods.


Fig. 12.19. Tree level ds → sd amplitude generated by Leff. Each of the two diagramshas a combinatoric factor of 2.

For simplicity, we shall discuss the procedure of large logarithm resummationin detail only for the top quark contribution, i.e. for the amplitude generated bythe diagrams in Fig. 12.17(a) with only top quarks in the electroweak loop, and byall the QCD corrections to this diagram. The fact of basic importance here is thatthis amplitude can be reproduced to all orders in QCD by the following effectivelagrangian (see Section 7.7):

Leff = LQCD(gluons and light quarks)

− G2F M2

W

4π2λ2

t C(µ)(�sγµPL�d

) (�sγ

µPL�d)

(12.148)

up to non-leading corrections in Q2low/(M2

W or m2t ) and higher order electroweak

corrections.The Wilson coefficient C(µ) depends on the scale µ at which the effective theory

fields and parameters are renormalized (see Section 7.7). The numerical value ofC(µ) can be found perturbatively in QCD by requiring equality of the effectivetheory and full Standard Model amplitudes up to O(external momenta/MW ). Atthe leading order in QCD, this means requiring equality of the box diagrams inFig. 12.17(a) (with only top quarks in the loop) and the tree level ds → sdamplitude generated by Leff (see Fig. 12.19). At the next-to leading order in QCD,we would need to take into account diagrams like the one shown in Fig. 12.18 onthe Standard Model side. On the effective theory side we would have to includeone-loop QCD corrections to the vertex generated by Leff (see Fig. 12.20).

At the leading order in QCD, the result we have already obtained in (12.146)immediately implies that

C(µ0) = S(xt , xt)+O(αs) (12.149)

Here, µ0 denotes the renormalization scale used in the effective theory when thecomparison of the full and the effective theory amplitudes is performed. At thefirst glance, it may seem strange that we specify this renormalization scale eventhough the tree level diagram in Fig. 12.19 obviously requires no renormalization.However, one needs to keep in mind that we intend to effectively resum large


Fig. 12.20. One-loop QCD corrections to the ds → sd amplitude generated by theeffective lagrangian. Each of the diagrams shown has its partner in which fermion lines inthe four-fermion vertex are differently contracted, and which enters with a relative minussign due to the Fermi statistics of quark fields, as in Fig. 12.19.

logarithms from all orders of the QCD perturbation series for the full StandardModel amplitudes. Thus, even if we restrict ourselves to the leading terms in theevaluation of C(µ0), we must make sure that no large logarithms reside in theneglected higher order QCD corrections to this coefficient coming from comparingamplitudes like the one in Fig. 12.18 with those in Fig. 12.20. It turns out that suchlarge logarithms are absent so long as the renormalization scale µ0 is chosen tobe of the same order as the heavy particle masses MW and mt in the full StandardModel diagrams.

Let us have a closer look at how the size of µ0 is being specified. The QCDcorrections to the tree level diagram, shown in Fig. 12.20, are UV-divergent. Afterrenormalization, they contain logarithms ln(µ2

0/Q2low). On the other hand, the box

diagram in Fig. 12.18 contains logarithms ln(M2W/Q2

low) and ln(m2t /Q2

low), i.e. thevery logarithms we would like to resum. The dependence on ln Q2

low cancels in thedifference of the full and effective theory amplitudes to all orders in QCD, up tonon-leading corrections in Q2

low/(M2W or m2

t ). (This is a general fact: Wilsoncoefficients are insensitive to low energy behaviour of the considered theory).Consequently, the QCD corrections to the leading order result for the coefficientC(µ0) depend on αs ln(M2

W/µ20) and αs ln(m2

t /µ20). In order to make the latter two

logarithms small (i.e. not much larger than unity), we choose µ0 of the same orderas MW and mt .


Once the Wilson coefficient C(µ) is found, we need to evaluate the K0 → K0

transition amplitude generated by Leff. This requires finding the matrix elementof the four-quark operator present in Leff between the kaon and antikaon states atrest. This matrix element is usually parametrized as follows (for states normalizedto unity): ∫

d3x 〈K0|(sLγµdL)(sLγµdL)|K0〉 = 1

3 BK(µ)mK f 2K (12.150)

where mK is the kaon mass, and fK � 161 MeV is its decay constant. Theparameter BK(µ) cannot be calculated perturbatively because it depends on QCDinteractions at energy scales close to �QCD.† Nevertheless, it is still a function ofthe renormalization scale µ used for UV renormalization of amplitudes generatedby the considered four-quark interaction. Non-perturbative methods used forevaluation of BK(µ) are not applicable when µ is arbitrarily large.‡ Consequently,renormalization group evolution of the Wilson coefficient from µ0 ∼ MW toµlow = (a few)×�QCD must be performed before calculation of the final low energyamplitude. Performing renormalization group evolution of the Wilson coefficientwould be necessary even if the matrix element was perturbatively calculable. Wewould then argue that the presence of large renormalization scale µ0 worsens theperturbative expansion of the matrix element due to large logarithms ln(µ2

0/Q2low).

Thus, the argument for using low µ in calculation of the matrix element would beidentical to the one we have used to fix the size of µ0.

The RGE for the Wilson coefficient C(µ) is discussed in Section 7.7. In orderto render the amplitudes generated by Leff finite, we need to perform the standardrenormalization of the LQCD part of it, as well as renormalize couplings and fieldsin the effective part:

C(µ) → ZcC(µ) (12.151)

and (�sγµPL�d

) (�sγ

µPL�d)→ Z2

2

(�sγµPL�d

) (�sγ

µPL�d)

(12.152)

The constant Z2 in the above equation is the usual QCD renormalization constant ofthe quark wave-function (see (8.4)). We recall the RGE for C(µ), (7.138), (7.139):(

µd

dµ− γc

)C(µ) = 0 (12.153)

† The factor BK(µ) is defined so that BK(µ) = 1 when only the vacuum state is introduced as an intermediatestate between the two currents in the matrix elements in (12.150), i.e. in the so-called vacuum saturationapproximation.

‡ For instance, maximal energy scales available in lattice calculations are much smaller than �QCD× (size ofthe lattice)/(lattice spacing).


where

γc = −µdZc

dµ

1

Zc. (12.154)

We are going to use this equation for evolving C(µ) from µ0 down to µlow at theleading order in QCD (i.e. in the so-called leading logarithmic approximation).Thus, we need to find the explicit O(αs) contribution to the renormalizationconstant Zc. At the leading order in QCD, the renormalization constant Zc inthe MS scheme is found from pole parts of one-loop diagrams shown in Fig. 12.20.Up to O(Q2

low/M2W ), the contribution to the effective lagrangian of Fig. 12.20(a),

(b) and the ‘negative partners’ of Fig. 12.20(c), (d) reads

L1 = 2 × (−X)(−igs)2(−i)i2µε

×∫

dnq

(2π)n

[�sγµPL(/q + md)γνT a�d][�sγνT a(/q + ms)γ

µPL�d]

(q2 − m2d)(q

2 − m2s )q

2

(12.155)

where X = G2F M2

Wλ2t C(µ)/4π2 abbreviates the factor standing in front of the

four-quark interaction in Leff. The SU (3) generators and gauge coupling aredenoted by T a and gs , respectively. As usually, colour indices are contractedwithin brackets. The factor of 2 is the combinatorial factor for the diagramssummarized by L1. It can be traced back to the possibility of two different (butindistinguishable) contractions of the fields in the four-fermion operator in (12.148)with the external fields after its insertion into (2.202) to build the Green’s function〈0|T (

�s�d�s�d) |0〉 (observe that for L1 all contributions are distinguishable and

see Fig. 12.19).By power counting, we can verify that only the double /q term in the numerator

can contribute to UV divergence. Thus, the only integral we need is∫dnq

(2π)n

qρqσ

(q2 − m2d)(q

2 − m2s )q

2= igρσ

32π2ε+ finite terms (12.156)

Substituting this result into the considered amplitude, one finds the pole part of L1

Lpole1 = − Xg2

s

16π2ε

(�sγµPLγργνT a�d

) (�sγ

νγ ργ µPLT a�d)

(12.157)

The pole parts of Fig. 12.20(c), (d) and the ‘negative partners’ of Fig. 12.20(a),(b) are found in an analogous manner and yield the contribution to the effectivelagrangian:

Lpole2 = − Xg2

s

16π2ε

(�sγνT aγργµPLγ

ργ νT a�d) (

�sγµPL�d

)


Pole parts of Fig. 12.20(e), ( f ) (and their ‘negative’ companions) are summarizedby

Lpole3 = + Xg2

s

32π2ε

(�sγνT aγργµPL�d

) (�sγ

νT aγ ργ µPL�d)

Lpole4 = + Xg2

s

32π2ε

(�sγµPLγργνT a�d

) (�sγ

µPLγργ νT a�d

)Compared with L1 and L2, L3 and L4 have no combinatorial factors of 2 since twoindistinguishable contractions appear when L3 (L4) is used in (2.202) (we recallonce again that for L1 and L2 all contractions are distinguishable and the factorof 2 is necessary to match the combinatorics of the tree level diagrams describedby the operator in (12.148), which gives pairs of indistinguishable contractions).Summing up all contributions one finds for the pole part

4∑k=1

Lpolek = − Xg2

s

16π2ε

[(�sγµγργν PLT a�d

) (�sγ

νγ ργ µPLT a�d)

+ (�sγνγργµγ

ργ ν PLT aT a�d) (

�sγµPL�d

)− (

�sγµγργν PLT a�d) (

�sγµγ ργ ν PLT a�d

)](12.158)

In order to simplify the Dirac algebra in the first expression, we apply theidentity (12.96). It holds in four dimensions only, but we are allowed to usefour-dimensional Dirac algebra so long as we are interested only in calculatingone-loop renormalization constants in the MS scheme (see also the discussion inSection 12.5 on muon decay). The second term in (12.158) is simplified by usingγνγργµγ

ργ ν = 4γµ +O(ε) and (T aT a)αβ = 43δαβ . After all that we find

4∑k=1

Lpolek = Xg2

s

16π2ε

[12(�sγ

µT a PL�d)(�sγ

µT a PL�d)

− 16

3

(�sγ

µPL�d)(�sγ

µPL�d)]

(12.159)

Finally, we can get rid of the SU (3) generators T a by applying the identity

(T a)αβ(Ta)γ δ = 1

2δαδδγβ − 16δαβδγ δ (12.160)

together with the Fierz rearrangement formula (12.98) for anticommuting fields.After all these simplifications, we obtain the final result

4∑k=1

Lpolek = − Xg2

s

12π2ε

(�sγ

µPL�d) (

�sγµPL�d

)(12.161)

with colour indices contracted within brackets. Now, we require that the divergent


part of the one-loop diagrams we have calculated is cancelled by the counterterm

Leff = −X (Zc Z22 − 1)

(�sγ

µPL�d) (

�sγµPL�d

)(12.162)

which implies that

Zc Z22 − 1 = − g2

s

12π2ε+O(g4

s ) (12.163)

Since the quark wave-function renormalization constant is known to be (seeChapter 8)

Z2 = 1 − g2s

6π2ε+O(g4

s ) (12.164)

the renormalization constant Zc must be equal to

Zc = 1 + g2s

4π2ε+O(g4

s ) = 1 + αs

πε+O(α2

s ). (12.165)

Consequently,

γc(µ) = αs(µ)

π+O(α2

s ). (12.166)

The solution to the RGE (12.153) is easily found to be

C(µ) = exp

{∫ lnµ

lnµ0

d(lnµ′) γc(µ′)}

C(µ0). (12.167)

Thus, we obtain (up to higher orders in αs)

C(µ) = exp

{1

π

∫ lnµ

lnµ0

d(lnµ′)αs(µ′)}

C(µ0)

= exp

{1

b1

∫ αs (µ)

αs (µ0)

dα′sα′s

}C(µ0) =

[αs(µ0)

αs(µ)

]−1/b1

C(µ0) (12.168)

where b1 is the coefficient in one-loop RGE for the QCD coupling constant (seeChapters 6, 8 and (6.45)):

µdαs

dµ= αs

[b1

αs

π+O(α2

s )]

(12.169)

The value of b1 = − 12(11 − 2

3 N f ) depends on the number N f of quark flavoursactive in the considered effective theory. In the full Standard Model, we have N f =6. In the effective theory introduced at the scale µ0, the top quark is already absent.Consequently, we substitute N f = 5 to b1 when running the Wilson coefficientfrom µ = µ0 to µ = µb ∼ mb. At this latter scale, the b quark should bedecoupled. Below µb, we substitute N f = 4 to b1. Finally, the charm quark can be


Fig. 12.21. Sample two-loop QCD corrections to the effective vertex.

decoupled at µ = µc ∼ mc. The ultimate expression for the Wilson coefficient atscales µlow < µc is thus

C(µlow) =[

αs(µc)

αs(µlow)

]6/27 [αs(µb)

αs(µc)

]6/25 [αs(µ0)

αs(µb)

]6/23

S(xt , xt) + O(αs).

(12.170)The expression we have obtained for C(µlow) is perturbative in αs but exact

in αs ln(µ0/µb), αs ln(µb/µc) and αs ln(µc/µlow). In other words, we haveresummed powers of αs×(large logarithm) from all orders of the usual perturbationseries. Achieving this resummation has been the main reason for introducing theeffective lagrangian (12.148).

Resummation of large logarithms can be systematically performed at higherorders in QCD, too. At the next-to-leading order, it requires calculating the O(αs)

part of C(µlow) exactly in αs ln(µ0/µb), αs ln(µb/µc) and αs ln(µc/µlow), i.e.summing all terms αs(αs ln(µ0/µb))

n etc. In order to achieve this, one needs toinclude O(αs) contributions to the matching between the full and effective theoriesat the scale µ0, and to the running of the Wilson coefficient down to the scale µlow.As mentioned earlier, next-to-leading contributions to the matching on the fulltheory side are given by two-loop diagrams like the one shown in Fig. 12.18. On theeffective theory side, one has to evaluate finite parts of the diagrams presented inFig. 12.20. Equating the two amplitudes renormalized at the scale µ0 ∼ MW , onefinds C(µ0) including the O(αs) part of it. Running the Wilson coefficient downto the low energy scale µlow at the next-to-leading order in QCD requires knowingthe two-loop contribution to the anomalous dimension γc. In the MS-scheme, it isgiven by pole parts of two-loop corrections to the effective four-quark vertex. Anexample of such a correction is shown in Fig. 12.21. Details of the next-to-leadingQCD calculation are presented in Buchalla, Buras & Harlander (1990).

Including the O(αs) part of C(µlow) in (12.170) automatically reduces itsdependence on the arbitrariness in the choice of scales µ0, µb and µc. Thesescales have to be of the same order of magnitude as MW , mb and mc, respectively.However, changing them by, for example, a factor of 2 is still allowed. For instance,when µ0 in (12.170) is replaced by 2µ0, the leading order (LO) contribution to the


coefficient C(µlow) changes by

CLO(µlow) =([

αs(2µ0)

αs(µ0)

]6/23

− 1

)CLO(µlow)

= −αs(µ0)

π(ln 2) · C(µlow)+O(α2

s ) (12.171)

Before the O(αs) term in (12.170) is found, the above dependence on µ0 can serveas an estimate of the error we make by neglecting this very term. On the other hand,when the O(αs) term is calculated explicitly, one finds that its µ0 dependence isprecisely such that it cancels the µ0 dependence of the leading term. Consequently,the µ0 dependence of C(µlow) becomes a O(α2

s ) effect (see, for example, Buras,Misiak, Munz & Pokorski (1995) for an extensive discussion of this point in thecase of the decay b → sγ ). The same phenomenon can be proven to happenat any order in perturbation theory. The renormalization-scale-dependence isalways formally a higher order effect. Thus, what matters in practical calculationsare only orders of magnitude of renormalization scales, not their precise values.Choosing the proper order of magnitude for the renormalization scale is importantfor (asymptotic) convergence of the perturbation series. On the other hand, theeffect of changing the scale by a factor of 2 can always be absorbed into unknownhigher order corrections.

In the remainder of this subsection, we shall briefly describe how the evaluatedWilson coefficient is used to find the C P-violating observable εK (defined inChapter 1). We recall that (1.273)

εK � Im(M12 A20)√

2 m|A0|2 exp(iπ/4). (12.172)

where M12 is the hermitean part of the weak hamiltonian, m is introduced in(1.273) and A0 is defined in (1.270). In order to simplify further discussion, weshall assume that we use the parametrization of the CKM matrix as in (12.44).When this parametrization is used, the amplitude A0 is approximately real. This isbecause the dominant contribution to A0 originates from the tree level s → uudweak transition. Therefore, A0 is to a good approximation proportional to V12V ∗

11

which is real in the standard parametrization (12.44) of the CKM matrix. Note thatif A0 was exactly real, then the parameters εK in (1.270) and ε in (1.265) would beequal. Our approximate expression for εK now becomes proportional to Im M12

εK � Im M12√2 m

exp(iπ/4) (12.173)

As follows from (1.261), the imaginary part of M12 is equal to

Im M12 = Im〈K0|HW|K0〉 +O(H 2W) = − Im〈K0|HW|K0〉 +O(H 2

W) (12.174)


The matrix element in the r.h.s. of (12.174) is generated by the box diagram inFig. 12.17 and by QCD corrections to it. Similarly to (12.146), we can write it as asum of contributions proportional to different products of CKM matrix elements

Im M12 = Im∑

i, j=u,c,t

λiλ j M (i j)12 (12.175)

The part proportional to Im λ2t in the above expression is given by the matrix

element of the effective four-quark interaction present in (12.148) between the kaonand antikaon states. Using (12.150) and (12.174), we find

Im M12 = −G2F M2

W

12π2Im(λ2

t )C(µlow)BK(µlow)mK f 2K

+ (contributions proportional to other products λiλ j ) (12.176)

with C(µlow) given by (12.170).Although the double top-quark contribution is really the largest in Im M12, the

other contributions cannot be neglected, because |λc| � |λt |. When the othercontributions are included and the unitarity of the CKM matrix is used, one obtainsa result which is usually written in the following form

Im M12 = −G2F M2

W

12π2BKmK f 2

K

{Im(λ2

c)S(xc)η1

+ Im(λ2t )S(xt)η2 + 2 Im(λtλc)S(xc, xt)η3

}(12.177)

where

S(x) = S(x, x)+ S(0, 0)− 2S(x, 0) = −3x3 ln x

2(1 − x)3+ x3 − 11x2 + 4x

4(1 − x)2

S(xc, xt) = S(xc, xt)+ S(0, 0)− S(xt , 0)− S(xc, 0)

= xc

[− ln xc + x2

t − 8xt + 4

4(1 − xt)2ln xt − 3xt

4(1 − xt)

]+O(x2

c )

(12.178)

and the µ-independent parameter BK is defined by

BK = BK(µ)[αs(µ)]−2/9[1 +O(αs)] (12.179)

The quantity η2 which parametrizes the short-distance QCD effects in the doubletop quark contribution is directly related to the calculated expression for thecoefficient C(µ). Namely

η2 = [αs(µc)]29

[αs(µb)

αs(µc)

]6/25 [αs(µ0)

αs(µb)

]6/23

+O(αs) (12.180)


The calculation of η1 and η3 is performed with the use of the same technique as η2,i.e. with help of an effective theory and renormalization group evolution of Wilsoncoefficients. However, it is technically much more involved.

Problems

12.1 Show that with N Higgs multiplets with weak isospin T 2n = tn(tn+1), n = 1, . . . , N ,

with vacuum expectation values vn , one gets the tree level relation:

ρ ≡ M2W

cos2 θW M2Z

=∑

n

[tn(tn + 1)− t2

3n

]v2

n

2∑

n t23nv

2n

(12.181)

12.2 Show that a convenient measure of the C P violation in the K0–K0 mixing is theparameter J = Im(VubVcd V �

ud V �cb). Show that it can be expressed in terms of four

moduli of the elements of the CKM matrix:

4J 2 = 4|Vub|2|Vcb|2|Vud |2|Vcd |2 (12.182)

− (1 − |Vud |2 − |Vcd |2 − |Vcb|2 − |Vub|2 + |Vud |2|Vcb|2 + |Vub|2|Vcd |2)2

12.3 Check that the relation ρ = 1 follows from the SUV(2) symmetry of the Higgspotential after spontaneous symmetry breaking (like in the σ model).

12.4 Calculate the β-functions for the renormalization group evolution of the gaugecouplings in the Standard Model.

12.5 Derive RGE for Yukawa couplings in the Standard Model.

12.6 Calculate the partial decay width for h0 → W+W−. Show that in the limit Mh �MW the same result is obtained by calculating the decay width for h0 → G+G−

where G± are the Goldstone bosons in the ’t Hooft–Feynman gauge. This is an exam-ple of the ‘equivalence theorem’: A(W±

L , Z0L, . . .) = A(G±, G0, . . .)+O(MW /E).

Show that the equivalence theorem also holds for decays h0 → W−W+ → 4 f andh0 → G+G− → 4 f (for MW < Mh < 2MW ), even in the limit of zero fermionmasses (i.e. for vanishing G± f f coupling), due to a singularity in the phase spaceintegration.

12.7 Show that �(h0 → γ γ ) ∼ (g2/M2W )α2 M3

h . Construct the effective lagrangian forthis process.

12.8 Calculate the forward–backward and left–right asymmetries for the process e+e− →Z0 → f f for s1/2 ≈ MZ .

12.9 Calculate the box diagram for K0–K0 mixing in the unitary gauge.

13

Chiral anomalies

13.1 Triangle diagram and different renormalization conditions

Introduction

Anomalies have already been mentioned in this book on several occasions. Inthis chapter we systematically discuss the fermion anomaly in (3+ 1)-dimensionalquantum field theory (Adler 1969, Bell & Jackiw 1969). Its existence can be tracedback to the short-distance singularity structure of products of local operators.

To be specific let us consider QCD. As discussed in Chapter 9, apart from beinginvariant under gauge transformations in the colour space its lagrangian is alsoinvariant under the global SU (N ) × SU (N ) × U (1) × UA(1) chiral group oftransformations acting in the flavour space.† Fermions belong to a vector-like, i.e.real, representation of the gauge group and for N > 2 to a complex representationof the chiral group (see Appendix E). We know from Chapters 9 and 10 that theNoether currents corresponding to the global flavour symmetry, although externalwith respect to the strong interaction gauge group, acquire important dynamicalsense. The axial non-abelian currents couple to Goldstone bosons (pseudoscalarmesons) and the left-handed chiral currents couple to the intermediate vectorbosons, i.e. they are gauge currents of the weak interaction gauge group. Moreover,the conservation of the U (1) current corresponds to baryon number conservationwhereas the conservation of the UA(1) current is a problem (the so-called UA(1)problem): it can be seen that the spontaneous breakdown of the SU (N )× SU (N )

implies the same for the UA(1) but there is no good candidate for the correspondingGoldstone boson in the particle spectrum.

It is clear from the preceding discussion that Green’s functions involvingchiral or axial currents are of direct physical interest. In the framework ofthe renormalization programme in QCD the finiteness of the matrix elements of

† We take quarks as massless. For massive quarks the chiral group is softly broken but, as we shall see, this isirrelevant for the discussion of anomalies.

457

458 13 Chiral anomalies

the T -ordered products of such local composite operators as chiral currents isan additional requirement which, in general, must be separately verified. Theymay require an additional renormalization. Anomalies reflect the absence of arenormalization procedure such that Green’s functions respect all the symmetriesof the lagrangian. In other words an anomaly means a deviation of the renormalizedtheory from the canonical behaviour.

Historically, the main tool for studying anomalies is the Ward identities inperturbation theory. Relations obtained in this way between various renormalizedGreen’s functions involving currents can then be summarized in operator language:certain currents have anomalous divergences. Ward identities, being a general-ization of Gauss’ theorem, also provide a beautiful manifestation of the UV–IRinterrelation in the theory: short-distance singularities manifest themselves in lowenergy theorems. In this way anomalies can be seen experimentally. However, ifgauge fields are coupled to linear combinations of vector and axial-vector flavourcurrents, as in the standard SU (2) × U (1) model of weak interactions, thesecombinations must be anomaly-free. This is an important constraint on any theory.Otherwise, the anomaly spoils the conservation of the currents coupled to gaugebosons and gauge theories with gauge fields coupled to non-conserved currents areinconsistent. In the next section we will see explicitly that the anomaly constitutesa breakdown of gauge invariance.

Anomalies originate in the short-distance singularities of the free fermionpropagator which generate renormalization ambiguities in the Green’s functionscontaining chiral currents. It turns out that in four space-time dimensions theGreen’s function which cannot be renormalized preserving all the classical sym-metries of the lagrangian is the one containing two vector and one axial currentsor three chiral currents. The basic structures to study are the triangle diagramsdepicted in Fig. 13.1. As we know, in perturbation theory, the requirement ofunitarity in each channel makes any one-loop amplitude well-defined up to apolynomial in the external momenta. This freedom is used in the renormalizationprocedure when we get rid of all infinite pieces. The triangle diagram is anomalousbecause it is impossible to add a polynomial so that all the classical symmetries aresimultaneously preserved.

In the following we calculate explicitly the triangle amplitude and study itsWard identities. All internal quantum numbers of fermions are suppressed in thiscalculation. We consider the U (1) and UA(1) currents of one-fermion free-fieldtheory. All the necessary generalizations can be easily introduced at the end.

To complete the introductory remarks it is worth mentioning that anomalies arealso of basic importance for most theories going beyond the Standard Model. Inparticular the problem of chiral anomalies in more than four space-time dimensionsand of gravitational anomalies as well as new techniques for their calculation

13.1 Triangle diagrams 459

Fig. 13.1.

have been vigorously studied (for example, Frampton & Kephart (1983), Alvarez-Gaume & Witten (1983), Zumino, Wu & Zee (1984)). We shall come back to someof these subjects later on.

Calculation of the triangle amplitude

We consider a vacuum matrix element of three chiral currents jLµ(x), say, or of one

axial vector current j5µ(x), and two vector currents jµ(x). For the time being we

treat them as abelian currents of the one-fermion free-field theory which, at theclassical level, are conserved. If we allow for a Dirac mass term in the lagrangianthen the axial-vector current is partially conserved

∂µ j5µ(x) = −2∂µ jL

µ(x) = 2mi�γ5�

Using the techniques of Section 10.1 and the fact that the canonical equal-time commutators [ j0(x, t), jµ(y, t)], [ j5

0 (x, t), jµ(y, t)] and [ j50 (x, t), j5

µ, (y, t)vanish one can derive the standard Ward identities for the matrix elements〈0|T jL

µ(z) jLν (y) jL

δ (x)|0〉 and 〈0|T j5µ(z) jν(y) jδ(x)|0〉. In momentum space, for

example, they read

−(p + q)µ�5µνδ(p, q) = pδ�5

µνδ(p, q) = qν�5µνδ(p, q) = 0 (13.1)

for m = 0 or, for m �= 0

−(p + q)µ�5µνδ(p, q) = 2m�5

νδ(p, q) (13.2)

and

pδ�5µνδ(p, q) = qν�5

µνδ(p, q) = 0 (13.3)


where p and q are momenta conjugate to x and y, respectively, �5µνδ(p, q) is the

Fourier transform of 〈0|T j5µ(z) jν(y) jδ(x)|0〉

�5µνδ(p, q)(2π)4δ(r + p + q)

=∫

d4x d4 y d4z exp[−i(px + qy + r z)]〈0|T j5µ(z) jν(y) jδ(x)|0〉 (13.4)

and �5νδ(p, q) is the Fourier transform of 〈0|T P(z) jν(y) jδ(x)|0〉 with P = �γ5�.

In a quantized theory, the current matrix elements get, at the one-loop level,a contribution from the triangle diagrams shown in Fig. 13.1 and from crosseddiagrams with the external lines interchanged, p ↔ q and δ ↔ ν. As we shall see,these contributions destroy the naive Ward identities (13.1)–(13.3).

We shall first consider the diagram on the r.h.s. of Fig. 13.1. The amplitudecorresponding to this diagram reads

Fµνδ(p, q) = (−1)∫

d4k

(2π)4Tr

[i

/k + /q − mγν

i

/k − mγδ

i

/k − /p − mγµγ5

](13.5)

and

�5µνδ(p, q) = Fµνδ(p, q)+ Fµδν(q, p) (13.6)

We take here a non-zero fermion mass but, as we shall see, the anomaly is mass-independent. Thus the result we will obtain is valid for the m = 0 case, too. Wealso keep in mind that, if currents are coupled to gauge fields with some couplingconstant g, an extra factor (−ig) appears in each such vertex.

The first observation worth making is the symmetry property

Fµνδ(p, q) = Fµδν(q, p) (13.7)

This follows from (13.5) by using the change of variables k ′ = −k, anticommutingγµγ5 = −γ5γµ and reversing the order of factors

Tr[γαγνγβγδγγ γµγ5] = Tr[γµγγ γδγβγνγαγ5] (13.8)

We also notice that the sign of the mass factor is irrelevant because the only non-vanishing mass-dependent terms in the trace are proportional to m2.

By power counting the amplitude Fµνδ(p, q) is superficially linearly divergent.It is impossible to renormalize this amplitude preserving simultaneously all threeidentities (13.1) or (13.2) and (13.3). It depends on our choice of the renormal-ization constraints, dictated by the physics of the problem, where the anomaly isactually placed.

For clarity of the following discussion it is useful to first write down the mostgeneral form of the amplitude �5

µνδ(p, q) = 2Fµνδ(p, q) consistent with the


requirements of parity and Lorentz invariance. It reads

�5µνδ(p, q) = A1(p, q)εµνδα pα + A2(p, q)εµνδαqα + B1(p, q)pνεµδαβ pαqβ

+ B2(p, q)qνεµδαβ pαqβ + B3(p, q)pδεµναβ pαqβ

+ B4(p, q)qδεµναβ pαqβ (13.9)

where As and Bs are scalar functions. The terms pνενδαβ pαqβ and qµενδαβ pαqβ

are not linearly independent. The requirement of Bose symmetry under interchangeof the two vector currents �5

µνδ(p, q) = �5µδν(q, p) implies that A1(p, q) =

−A2(q, p), B1(p, q) = −B4(q, p), B2(p, q) = −B3(q, p). Using again powercounting arguments it is clear that only the functions A1 and A2 are divergent,in fact only logarithmically divergent, whereas the functions Bi are finite. Sub-tractions are, in principle, necessary for the functions A1 and A2. However,these functions can be expressed in terms of Bs when definite renormalizationconditions are imposed. Imagine, for instance, that we insist on the vector currentconservation

qν�5µνδ(p, q) = pδ�5

µνδ(p, q) = 0 (13.10)

for the renormalized vertex �5µνδ. Then we must have

A1 + p · q B1 + q2 B2 = 0

A2 + p2 B3 + p · q B4 = 0

}(13.11)

and indeed the conditions (13.11) determine the divergent amplitudes Ai in termsof finite Bi s. From Bose symmetry A1(p, q) = −A2(q, p) follows so that onlyone of the equations (13.11) is independent.

There are many ways to calculate anomalies. One is to regularize the amplitude(13.5) by some regularization prescription and to calculate Ward identities for theregularized �5

µνδ. If the regularization procedure satisfies the constraints which wewant to impose as our renormalization constraints then in the limit of no regulatorwe obtain the anomalous Ward identity we are looking for. Given this identity, byadditional finite subtractions one can always transform it into another anomalousWard identity, satisfying different renormalization constraints. Let us follow thisapproach choosing to work with the Pauli–Villars regularization (Itzykson & Zuber1980) of the amplitude (13.5). Dimensional regularization in this case has someproblems with ambiguities in the extension of γ5 to n dimensions (for a review ofthis matter see, for instance, Ovrut (1983) and references therein). Let us chooseas our renormalization conditions the relations (13.10). Thus we insist on vectorcurrent conservation for the renormalized vertex �5

µνδ

The Pauli–Villars regularization of the amplitude �5µνδ(p, q) consists of con-

sidering it to be a function of the mass of the fermion circulating in the loop:


�5µνδ(p, q) = �5

µνδ(p, q,m). A regulated amplitude is defined as the differencebetween the given amplitude and the same amplitude taken at some other value Mof the mass (M is the regulator)

�5µνδ(p, q,m, M) = �5

µνδ(p, q,m)− �5µνδ(p, q, M) (13.12)

The regularized amplitude �5µνδ(p, q,m, M) is finite. This can easily be checked

by expanding the integrand in the integrals over d4k in powers of 1/k2: the lineardivergence of �5

µνδ(p, q,m) is cancelled by �5µνδ(p, q, M). At the end one lets the

mass M go to infinity and the final answer is finite with no additional subtractions,because it satisfies our renormalization conditions fixing Ai in terms of Bi

First we show that the regularized amplitude satisfies the ‘normal’ Wardidentities

−(p + q)µ�5µνδ(p, q,m, M) = 2m�5

νδ(p, q,m)− 2M�5νδ(p, q, M) (13.13)

pδ�5µνδ(p, q,m, M) = qν�5

µνδ(p, q,m, M) = 0 (13.14)

Indeed, using the relation /p+ /q = (/q + /k −m)− (/k + /p−m) we can write downthe following equation:

−(p + q)µ�5µνδ(p, q,m, M)

= 2i∫

d4k

(2π)4Tr

[1

/k + /q − mγν

1

/k − mγδγ5 − (m → M)

]+ 2i

∫d4k

(2π)4Tr

[γν

1

/k − mγδ

1

/k − /p − mγ5 − (m → M)

]+ 2i

∫d4k

(2π)4Tr

[1

/k + /q − mγν

1

/k − mγδ

1

/k − /p − m2mγ5 − (m → M)

](13.15)

All the integrals are finite. The first two terms are second rank pseudotensorsdepending on only one momentum variable and hence vanish. The last term givesjust the Ward identity (13.13). Consider now pδ�5

µνδ(p, q,m, M). Using /p =(/k − m)− (/k − /p − m) this becomes

pδ�5µνδ(p, q,m, M)

= 2i∫

d4k

(2π)4Tr

[1

/k + /q − mγν

1

/k − /p − mγµγ5 − (m → M)

]− 2i

∫d4k

(2π)4Tr

[1

/k + /q − mγν

1

/k − mγµγ5 − (m → M)

](13.16)

The last term vanishes for the same reason as above. The first term also vanishesbecause we can shift the momentum integration variable to k ′ = k − p and


again apply the same argument. This shift of the integration variable is legitimatebecause the integral is finite. If it were linearly divergent the translation of theintegration variable might involve a non-vanishing surface term which would spoilthe argument† (see Problem 13.1). In a quite analogous way we can show that

qν�5µνδ(p, q,m, M) = 0

Thus, our regularization procedure is consistent with our renormalization require-ments (13.10). Our last step is to take the limit M → ∞ in the Ward identity(13.13). The anomaly results from the non-vanishing of the regulator term in(13.13) in that limit. The �5

νδ(p, q,m) vertex can be calculated explicitly sincethe Feynman integral is convergent:

�5νδ(p, q,m) = 2i∫

d4k

(2π)4Tr

[1

/k + /q − mγ ν 1

/k − mγ δ 1

/k − /p − mγ5

]= 2 × 4miενδβα pβqα F(p, q,m) (13.17)

where

F(p, q,m) = i∫

d4k

(2π)4

1

(k + q)2 − m2

1

k2 − m2

1

(p − k)2 − m2(13.18)

and the factor 2 in front of the integral in (13.17) takes account of the crosseddiagram.

Using the Feynman parametrization (B.16) and integrating over d4k one gets

F(p, q,m) = − 1

16π2

∫ 1

0dx∫ 1−x

0dy [q2 y(1− y)+ p2(1− x)x+2pqxy−m2]−1

(13.19)It is now straightforward to calculate the limit

limM→∞

2M�5νδ(p, q, M) = +(i/2π2)ενδβα pβqα (13.20)

and the final form of the axial Ward identity consistent with our renormalizationconstraints reads

−(p + q)µ�5µνδ(p, q,m) = 2m�5

νδ(p, q,m)− (i/2π2)ενδβα pβqα

pδ�5µνδ = qν�5

µνδ = 0

}(13.21)

The anomalous term is m-independent.Another way to obtain the anomalous Ward identity (13.21) is to calculate

explicitly the finite functions Bi starting with the unregularized amplitude (13.5)and then to use renormalization constraints (13.11). This we leave as an exercisefor the reader.† Note the difference between a shift of the integration variable by a constant and a change of variables k →−k

used in proving (13.7).


Different renormalization constraints for the triangle amplitude

The renormalization conditions (13.10) lead to the result (13.21). Whether weshould impose them or not depends on the physics of the problem. This will beillustrated in the next section. It may happen that we must, instead, insist on therelation

−(p + q)µ�5µνδ(p, q,m) = 2m�5

νδ(p, q,m) (13.22)

as our renormalization constraint. The divergent amplitudes A1 and A2 are againuniquely specified by the condition (13.22). Indeed, with an additional finitesubtraction for the A1 and A2

a1εµνδα pα + a2εµνδαqα (13.23)

we see, by comparing (13.21) and (13.22), that the latter is satisfied for

a1 − a2 = i/2π2

Since a1(p, q) = −a2(q, p) from the Bose symmetry under interchange of the twovector currents discussed earlier, we conclude that

a1 = −a2 = i/4π2 (13.24)

Thus, with (13.22) as the renormalization condition one gets

pδ�5µνδ(p, q,m) = −(i/4π2)εµναβ pαqβ

qν�5µνδ(p, q,m) = (i/4π2)εµδαβ pαqβ

}(13.25)

One cannot satisfy simultaneously all three Ward identities (13.2) and (13.3).Finally let us consider the triangle on the l.h.s. of Fig. 13.1 with the left-handed

currents at each vertex and in the limit m = 0. At the level of the divergentFeynman integral the bare amplitude �L

µνδ(p, q) formally satisfies the followingrelation

�Lµνδ(p, q) = 1

2�µνδ(p, q)− 12�

5µνδ(p, q) (13.26)

where �µνδ(p, q) is the three-vector current amplitude which has no anomaly.Thus formally the anomaly of �L

µνδ(p, q) can also be calculated from the triangleon the r.h.s. of Fig. 13.1. However, our previous renormalization conditions areinappropriate since now for the Green’s function of three identical currents wemust require Bose symmetry of the renormalized �5

µνδ(p, q) (which we denote

by �5µνδ(p, q) to distinguish it from the AVV amplitude considered earlier) under

interchange of any pair of vertices. We can get the right answer by applying afinite renormalization to the previous result (13.21). Adding a finite polynomial


term, (13.23), to the amplitudes A1 and A2 we get (for m = 0)

−(p + q)µ�5µνδ(p, q) = (−i/2π2 + a1 − a2)ενδαβ pαqβ

pδ�5µνδ(p, q) = a2εµναβ pαqβ

qν�5µνδ(p, q) = a1εµδαβ pαqβ

(13.27)

Insisting on symmetry under the interchange −(p + q) ↔ p, µ ↔ δ and p ↔q, δ ↔ ν one gets the following equations

−i/2π2 + a1 − a2 = a2

a1 = −a2

}(13.28)

Therefore

a1 = −a2 = i/6π2 (13.29)

and

−(p + q)µ�5µνδ(p, q) = −(i/6π2)ενδαβ pαqβ

pδ�5µνδ(p, q) = −(i/6π2)εµναβ pαqβ

qν�5µνδ(p, q) = +(i/6π2)εµδαβ pαqβ

(13.30)

Thus, the same triangle amplitude leads to three different sets of anomalous Wardidentities depending on the imposed renormalization conditions. The latter must bechosen according to the physical requirements of the problem under consideration.We recall that our present interest is in the amplitude �L

µνδ(p, q) so we combine(13.30) with (13.26).

Important comments

In the next section we will discuss in more detail the anomaly problem in thecontext of specific physical theories. Here, to illustrate the meaning of arbitrarinessin renormalization conditions for the triangle diagram, we consider two modelsgiven by the lagrangians

L = Z2�iD/� − 14 Z3 Fµν Fµν (13.31)

and

L = ZR2 �Ri/∂�R + ZL

2 �LiD/�L − 14 Z3 Fµν Fµν (13.32)

respectively, where

Dµ = ∂µ + ig Aµ, Fµν = ∂µ Aν − ∂ν Aµ


Fig. 13.2.

and the lagrangians are written in terms of renormalized quantities. We define thebare currents, for instance vector and axial-vector currents in the theory (13.31),which in terms of the renormalized fields are as follows:

j5µ(x) = Z2�(x)γµγ5�(x) (13.33)

and

jµ(x) = Z2�(x)γµ�(x) (13.34)

A comment is in order here on the renormalization of currents in presence ofthe axial anomaly. In Section 10.1 we have proved that a conserved or partiallyconserved current satisfying normal Ward identities is not renormalized. Thisconclusion is no longer valid in the presence of the anomaly. The current is stillmultiplicatively renormalized R j5

µ = Z5 j5µ, but Z5 is not finite (we will come

back to this point shortly). However, as we know from the previous subsection,at the one-loop level the current matrix elements are finite after imposing certainrenormalization conditions, without additional counterterms. Thus, at this levelZ5 = 1.

The first theory is just QED of a single fermion field. The vector current iscoupled to the electromagnetic field and the axial-vector current, conserved in theNoether sense, is not coupled to any gauge field. Gauge invariance of the theoryrequires the vector current conservation. Thus for the triangle diagram involvingj5µ and two vector currents we must impose the renormalization conditions (13.10)

which lead to the anomalous Ward identity (13.21). In operator language thiscorresponds to an anomaly in the divergence of the axial-vector current

∂µ j5µ(x) = (g2/16π2)εµνρδ Fµν(x)Fρδ(x) (13.35)

Eq. (13.35) can be verified using the Feynman rules for the operator insertion

εµνρδ Fµν(x)Fρδ(x) = 4εµνρδ∂µ Aν(x)∂ρ Aδ(x) (13.36)

(see Section 7.6). Considering, for example, the Green’s function 〈0|T ∂µ Aν(z)


Fig. 13.3.

Fig. 13.4.

∂ρ Aδ(z)Aα(y)Aβ(x)|0〉 in momentum space one gets the Feynman diagram shownin Fig. 13.2 (propagators are, as usual, removed from the external lines). Therefore,in order g2 the Fourier transform (13.21)† (with m = 0) of the divergence∂µ〈0|T j5

µ jν jδ|0〉 is given by the Feynman diagram in Fig. 13.2 and (13.35) followsif we recall the general form (10.11) of Ward identities. Using (13.35) we cancalculate perturbatively other anomalous Ward identities involving axial-vectorcurrent j5

µ(x). For instance, the derivative of the Green’s function �5µ��

=〈0|T�(x) j5µ(0)�(y)|0〉 is anomalous because the latter gets a contribution fromthe diagram in Fig. 13.3 and in the lowest non-vanishing order in g2 the anomalyis given by the diagram in Fig. 13.4. The anomalous Ward identity for �5

��(derive

it using the path integral technique of Sections 10.2 and 13.3) can be used to studythe renormalization properties of the axial-vector current. Closely following theprocedure of Section 10.1 we conclude that Z5 is not finite as the operator Fµν Fµν

does not have finite matrix elements: e.g. the diagram in Fig. 13.4 is logarithmicallydivergent.

Our second example, the lagrangian (13.32), is a theory in which only theleft-handed chiral current couples to the gauge fields. The triangle diagram withgauge source currents at each vertex should be evaluated imposing symmetry underinterchange of any pair of indices. Using relations (13.26) and (13.30) we now get

† Now with (−ig) at each vector vertex included.


the anomaly in the chiral current

∂µ jLµ(x) = −(g2/48π2)Fµν Fµν (13.37)

Because of the anomaly the gauge source current is no longer conserved and thetheory is not gauge invariant.

Writing the operator equation (13.35) for the bare current we have used theresult obtained in the lowest order in g2 in perturbation theory. Equivalentlyone can say that the gauge field has been treated as a classical background field.One believes that the anomalies are not modified by higher order corrections,i.e. (13.35) is exact to any order in g2 for the bare current in the regularizedtheory or for the renormalized operators (see also the next section) with g andFµν being the renormalized quantities defined by the lagrangian (13.31). Forthe vector abelian theory considered here this has been proved to all orders inperturbation theory (Adler & Bardeen 1969). The proof is technical but its mainpoint can be summarized as follows: the necessity of renormalization of thehigher order corrections in g2, i.e. of photon insertions in the triangle diagram,does not interfere with chiral symmetry, so the chiral anomaly should arise onlyfrom the lowest order fermion loops. (On the contrary, the scale invarianceanomaly discussed in Chapter 7 does obtain corrections because renormalizationviolates scale invariance.) The Adler–Bardeen theorem also holds in non-abeliangauge theory but it may be regularization-scheme-dependent (Bardeen 1972),particularly in supersymmetric theories. In gauge theories with chiral gaugesymmetry the anomalies must be absent for a consistent theory. Accordingto the Adler–Bardeen theorem generalized to the non-abelian case a theory iscompletely free of anomalies if, and only if, all the triangle diagram anomaliesare absent.

Our last remark in this section is concerned with generalizing the trianglediagram calculation to the case in which several species of fermion fields whichhave some internal degrees of freedom are present. Let the currents in the verticeshave coupling matrices λa

1, λb2, λ

c3. The currents may be coupled to gauge fields or

not and the matrices λa1, λ

b2, λ

c3 may transform non-trivially under different internal

symmetry groups and be block-diagonal in the remaining internal symmetryspaces. For the triangle diagram we get then an extra factor 1

2 Tr[λa1{λb

2, λc3}],

where the trace corresponds to summing over different fermions in the loop andthe anticommutator is due to the symmetrization over the two diagrams in Fig. 13.5The necessary modifications of operator equations, like (13.35), will be discussedlater.

13.2 Physical consequences of the chiral anomalies 469

Fig. 13.5.

13.2 Some physical consequences of the chiral anomalies

Chiral invariance in spinor electrodynamics

The lagrangian (13.31) of spinor electrodynamics is invariant both under U (1) andaxial UA(1) transformations. However, in quantum theory if we want to preservevector gauge invariance an anomaly appears in the divergence of the axial-vectorcurrent. If we introduce a cut-off which preserves photon gauge invariance thento any order in g2 the divergence of the unrenormalized axial-vector current isgiven by (13.35). As discussed in the previous section, the triangle diagramamplitude turns out to be finite even in the limit of no cut-off but the higher ordercorrections induce infinities in the axial-current matrix elements. Not all of thesecan be absorbed in the renormalization constants present in the lagrangian (13.31).Therefore due to the anomaly the axial-vector current is renormalized:

R j5µ = Z5 j5

µ

and we must re-express (13.35) in terms of the renormalized operators. Defining

(g2/8π2)[Fµν(x)Fµν(x)]R = (g2/8π2)Fµν(x)Fµν(x)+ (Z5 − 1)∂µ j5µ(x)

(13.38)

we may write (13.35) in terms of the renormalized operators

∂µ R j5µ(x) = (g2/8π2)[Fµν(x)Fµν(x)]R (13.39)

However, this renormalized axial-vector current does not generate chiral trans-formations. The axial charge is not conserved, it does not commute with thephoton field and does not have the correct equal-time commutation relations withthe renormalized spinor field, due to the presence of a dimension four operator(Fµν Fµν)R in the divergence of the current (see Section 10.1).

Nevertheless, it turns out that chiral invariance remains a symmetry of thequantum theory based on the lagrangian (13.31) and it is possible to constructthe corresponding charge Q5. Following Adler (1970) we first define a new bare


current: symmetry current

jS5µ(x) = j5

µ(x)− (g2/4π2)Aν(x)Fµν(x) (13.40)

which, from (13.35), is conserved

∂µ jS5µ = 0 (13.41)

Its existence is directly related to our discussion of different renormalizationconditions of the triangle diagram; compare (13.21) and (13.22). This current is afinite operator (Bardeen 1974) and does not need renormalization. The current jS

5µ

is conserved but it is explicitly gauge-dependent due to the presence of the gaugefield in its definition and therefore is not an observable operator. But the associatecharge

Q5(t) =∫

d3x jS50(x, t) (13.42)

is from (13.41) time-independent, its commutator with �(x), calculated by use ofthe canonical commutation relations, is

[Q5(t),�(x, t)] = −γ5�(x) (13.43)

it commutes with the photon field and it is gauge-invariant. Let us check the lastproperty. Under the gauge transformation

δ�(x) = −ig�(x)�(x)

δAµ(x) = ∂µ�(x)

}(13.44)

the symmetry current transforms as follows:

δ jS5µ = (g2/4π2)∂ν�(x)Fµν(x) = (g2/4π2)∂ν(�(x)Fµν(x)) (13.45)

Therefore

δQ5(t) =∫

d3x δ jS50(x) ∼

∫d3x ∂ν(�(x)F0ν(x)) =

∫d3x ∂k(�(x)F0k(x)) = 0

(13.46)Gauge invariance of Q5 in spinor electrodynamics follows from the vanishingof �(x)F0k(x) at spatial infinity. The situation is different in non-abelian gaugetheories (see Section 8.3) where not all finite gauge transformations can be reachedby iterating infinitesimal transformations. We conclude that in spite of the presenceof the axial anomaly chiral symmetry remains a good symmetry in massless spinorelectrodynamics.


πππ0 → 2γγγ

The relevance of the chiral anomaly to the decay π0 → 2γ relies on our iden-tification of pions as the Goldstone bosons of the spontaneously broken chiralsymmetry of QCD (see Chapter 9). Thus, the axial-vector flavour currents of theSU (2)×SU (2) symmetric QCD lagrangian (9.1) acquire dynamical sense throughtheir coupling to pions, see (9.65)

〈0| j5aµ (x)|πb〉 = i fπ pµδ

ab exp(−ipx)

where pµ is the pion four-momentum. To calculate the π0 → 2γ decay rate weconsider the amplitude ( j5a

µ (x) is the renormalized axial isospin current)∫d4x exp(−iqx)〈γ1(k1)γ2(k2)|T Sj53

µ (x)|0〉 =∫

d4x d4 y d4z exp(+ik1 y)

× exp(+ik2z) exp(−iqx)〈0|T j53µ (x) jν(y) jδ(z)|0〉εν1(k1)ε

δ2(k2) (13.47)

In (13.47) we have used (2.188), (2.68) and the equations of motion for the gaugefields Aµ(x). Next, we use the PCAC (partial conservation of the axial-vectorcurrent) approximation relating physical quantities to the results obtained in theexact symmetry limit p2 = m2

π = 0. In this limit there is a pion pole contributionto the l.h.s. of (13.47) (see, for example, (2.194), (2.195) and (2.196))

limq2→p2

∫d4x exp(−iqx)〈γ1(k1)γ2(k2)|T Sj53

µ (x)|0〉 = fπ(pµ/p2)〈γ1γ2|S|π0〉+ O(k1, k2) (13.48)

where O(k1, k2) has no pole in p2.We now take the divergence of (13.47). The divergence of the r.h.s. of (13.47)

is given by the results of the previous section with the triangle diagram specifiedas shown in Fig. 13.6. The fermion lines are quark lines, the Pauli matrix σ 3 is theneutral axial-vector coupling matrix and Q is the diagonal quark electric chargematrix. In this case, for �5

µνδ defined by (13.4) we get the following:

pµ�5µνδ(k1, k2) = (i/2π2)Tr[ 1

2σ3 Q2]ενδβαkβ

1 kα2 (13.49)

plus terms of at least third order in ki from the second diagram: the factor k1k2

comes from the gauge-invariant coupling (through the field strength tensor Fµν) ofexternal photons and an extra factor (k1+k2) comes from the anomalous divergenceterm. The effect of the last diagram is only the coupling-constant renormalization.With u and d quarks of three colours we get

Tr[ 12σ

3 Q2] = 3 × 12(

49 − 1

9)e2 = 1

2 e2 (13.50)


Fig. 13.6.

Fig. 13.7.

Therefore, from (13.47) and (13.48)

〈γ1γ2|S|π0〉 = 1

fπ

e2

4π2iενδβαkβ

1 kα2 ε

ν1ε

δ2(2π)4∂(p − k1 − k2)+ O(k1, k2, p)

(13.51)

Our calculation of the divergence of (13.47) is, of course, equivalent to calculatingperturbatively the matrix element 〈γ1γ2|T S∂µ j5a

µ (x)|0〉 with the divergence givenby

∂µ j5aµ (x) = (e2/8π2) 1

2 Tr[ 12σ

a{Q, Q}]Fµν Fµν (13.52)

The contributing diagrams are those shown in Fig. 13.7. Terms O(k1, k2) in(13.48) and O(k1k2, p) in (13.51) are at least second and third order in momenta,respectively.

From Lorentz and parity invariance

〈γ1γ2|S|π0〉 = f (p2)iενδβαkβ

1 kα2 ε

ν1ε

δ2(2π)4δ(p − k1 − k2) (13.53)


Fig. 13.8.

Comparing (13.51) and the last equation we conclude that

f (0) = 1

fπ

(e2

4π2

)= α

fππ(13.54)

and the rate for π0 → 2γ is

� = 1

2mπ

∑ε1,ε2

∫d3k1 d3k2

2π64k10k20| f (0)ενδβαkβ

1 kα2 ε

ν1ε

δ2|2(2π)4δ(p − k1 − k2)

= α2m3π

64π3 f 2π

= 7.63 eV (13.55)

(Bose statistics implies integration over a 2π solid angle only). The experimentalvalue is �Exp = (7.7 ± 0.5 eV). We make two important observations. The result(13.55) is entirely given by the axial anomaly. In the absence of the anomaly onewould get f (0) = 0 and PCAC could not be reconciled with the π0 → 2γ decayrate. Secondly, the agreement with the experimental value relies on the existenceof three colour states of quarks.

Chiral anomaly for the axial U (1) current in QCD; UA(1) problem

Global symmetries of the QCD lagrangian with massless quarks are (see Sec-tion 9.1): chiral SU (n) × SU (n) invariance and invariance under abelian vectorU (1) and axial-vector UA(1) transformation. We study now the anomaly of theUA(1) current. The UA(1) current is a singlet under the colour group and therelevant triangle diagram is shown in Fig. 13.8. We must have Dµ j a

µ = 0 tomaintain gauge invariance. A derivation of the equation Dµ j a

µ = 0 in a quantizedgauge-invariant theory is given in the next section. For the triangle diagramwe effectively get the condition ∂ν〈0|T j5

µ j aν j b

δ |0〉 = 0 and therefore we mustimpose the renormalization constraints (13.10). Thus, by analogy with (13.21),the divergence −(p+ q)�5

µνδ(p, q) of the Fourier transform of the matrix element


〈0|T j5µ j a

ν j bδ |0〉 reads

−(p + q)µ�5µνδ(p, q) = (ig2/2π2)n Tr[ 1

2λa 1

2λb]ενδβα pαqβ (13.56)

where n is the number of quark flavours. The operator equation (13.35) can alsobe easily generalized to the present case if we remember that UA(1) current is acolour singlet. The only colour-invariant operator which for the triangle diagramgives the result (13.56) is

∂µ j5µ(x) = −(n/8π2)Tr[GµνGµν] (13.57)

where Tr is in the internal quantum number space and Gµν = igGaµνT a, T a = 1

2λa .

A new feature of (13.57) is that, due to the self-coupling of the non-abelian gaugefields, see (1.111), the anomalous divergence of the UA(1) current involves notonly terms quadratic in gauge fields but also terms of the third order in gauge fields

∂µ j5µ(x) = −(n/16π2)Tr[4∂µ Aν∂ρ Aσ + 4∂µ Aν[Aρ, Aσ ]]εµνρσ

= −(n/4π2)εµνρσ ∂µ Tr[Aν∂ρ Aσ + 23 Aν Aρ Aσ ] (13.58)

Thus, square diagrams also contribute to the anomaly for the divergence of the axialcurrent. However, this contribution is not independent of the triangle anomaly:given the latter, it is determined by the gauge invariance of the axial-vector current.All the anomalous Ward identities follow from the operator equation (13.57). Foran explicit discussion of the square diagrams in the spirit of the previous sectionsee, for instance, Aviv & Zee (1972).

The anomaly (13.57) of the axial-vector current is crucial for resolving theso-called UA(1) problem in QCD. As we know from Chapter 9 the massless QCDlagrangian has the chiral (U (1)∓UA(1)) symmetry. As with the SU (n)× SU (n)it does not seem to be the symmetry of the hadron spectra so UA(1) must bebroken (U (1) manifests itself in baryon number conservation). However, unlikethe case of the spontaneously broken SU (n) × SU (n), there is no good candidatefor the UA(1) Goldstone boson in the particle spectrum. The resolution of theproblem is in the existence of the anomaly (13.57) and of field configurationswith a non-zero topological charge (see Section 8.3). These properties offer amechanism of breaking of the UA(1) which does not imply the existence of themassless Goldstone boson coupled to the gauge-invariant UA(1) current.

As in the first subsection of this section we observe that the very existence of theanomaly does not destroy the UA(1) symmetry since the conserved current

jS5µ = j5µ − 2nKµ, ∂µKµ = −(1/16π2)Tr[GµνGµν]

and the conserved charge

Q5 =∫

d3x jS50


can be defined. Now, however, Q5 is not gauge-invariant: it is not gauge-invariantunder large gauge transformations GN with the winding number N . We have (seeSection 8.3 and in particular (8.65) and (8.67) and Problem 8.2)

GN Q5G−1N = Q5 − 2nN

From Section 8.3 we also remember that

GN |�〉 = exp(−iN�)|�〉 (8.58)

Thus we get

GN exp(i�′Q5)|�〉 = exp[−iN (�+ 2n�′)] exp(i�′Q5)|�〉or

exp(i�′Q5)|�〉 = |�+ 2n�′〉The conclusion is that the UA(1) rotation by an angle �′ changes the |�〉 vacuuminto the |� + 2n�′〉 vacuum. UA(1) symmetry is spontaneously broken becausethe vacuum is not UA(1)-invariant. Different �-vacua are degenerate states sincefor massless fermions H commutes with Q5. However, it can be seen fromthe chiral Ward identities that the spontaneous breakdown occurring because ofthe axial-vector anomaly and the existence of field configurations with non-zerotopological charge (hence the set of the �-vacua) is consistent with the assumptionof no Goldstone boson coupled to the gauge-invariant current j5

µ.

Anomaly cancellation in the SU (2)×U (1) electroweak theory

We discuss this problem at the level of the triangle diagrams relegating to the nextsection the construction of the full anomalous divergence of a non-abelian currentcoupled to gauge fields. This does not limit the generality of our discussion: atheory is anomaly-free if, and only if, the triangle diagram anomalies are absent.For consistency of a quantum gauge theory the gauge source currents must becovariantly conserved, i.e. anomaly-free (Gross & Jackiw 1972; see also Jackiw1984 and our discussion in the next section).

In the electroweak theory the SU (2) gauge bosons couple only to the left-handedcurrents and the U (1) gauge boson couples to both the left- and right-handedcurrents but with different U (1) charges Y . It will also be convenient to rememberthat by construction the fermion electric charge operator is given by Q = T 3 + Y .Thus, YR = Q and YL = (Q − T 3). We can now systematically list possibleanomalous triangle diagram contributions to the divergences of the gauge sourcecurrents. Take first the SU (2) current j a

Lµ: we can have the triangle diagramsshown in Fig. 13.9 with the j a

Lµ in one of the vertices and the corresponding


Fig. 13.9.

symmetrized diagrams ( jYLµ is the left-handed part of the U (1) current). It is

clear from our discussion in the previous section that the eventual anomalouscontribution of these diagrams after summing over all fermions in the loops isproportional to

12 Tr[T a{T b, T c}] for Fig. 13.9(a)12 Tr[T a{Y, Y }] for Fig. 13.9(b)12 Tr[T a{Y, T b}] for Fig. 13.9(c)

(T a = σ a are SU (2) generators). We immediately see that Fig. 13.9(b) does notcontribute because [Y, T a] = 0 and Tr T a = 0. If we express Y in terms of Q theremaining contributions are

12 Tr[T a{Q, T b}] = 1

2 Tr[Q{T b, T a}] = δab Tr[Q(T a)2] = 12δ

ab∑

i

Qi (13.59)

where the sum is taken over the charges of all fermions in the loop, and

Tr[T a{T b, T c}] = 0

for the group SU (2).The anomalous contribution to the divergence of the U (1) current jY

µ =∑i (�

iLγµY� i

L + � iRγµY� i

R), where the sum is over all fermions in the theory,can be studied in a similar way. We recall that the U (1) coupling is not vector-likebecause of different Y quantum number assignments to the left- and right-handedfields. In this case all possible triangle diagrams are proportional to one of thefollowing factors:

12 Tr[Y {Y, T a}], 1

2 Tr[Y {T a, T b}] and 12 Tr[Y {Y, Y }]

The first two traces have already been calculated. The last one appears in two typesof loop, with left- and right-handed fermions, which contribute with opposite signsbecause of the (1 ± γ5) factors (see (13.26)). We calculate

12 Tr[Y {Y, Y }] = Tr Y 3 = Tr(Q − T3)

3 = Tr[Q3 − 3Q2T3 + 3Q(T3)2 − (T3)

3]

(13.60)


Since Q is the same for fermions with either of the chiralities, the Q3 term iscancelled separately for each fermion when the sum over the two types of loop istaken. The remaining terms give again∑

i

Qi = 0 (13.61)

since Tr[Q2T 3] = Tr[(T3+Y )2T3] = Tr[(T3)3+2Y (T3)

2+Y 2T3] = Tr Y = Tr Q.Thus (13.61) is the only condition for the anomaly cancellation. We conclude thatthe SU (2) × U (1) theory is anomaly-free if the sum of charges of all fermions inthe theory vanishes. The quarks and leptons discovered so far strikingly satisfythis condition: to each SU (2) doublet of leptons corresponds a doublet of quarks,fractionally charged but coming in three colours.

An interesting point to mention in the context of the SU (2)×U (1) theory is thatonce we insist on conservation of the chiral source currents for the electroweakgauge fields (one needs suitable sets of fermion multiplets to ensure this property)the baryon number current, U (1) vector current, acquires an anomaly. This is clearif we consider the triangle diagram with the baryon current and two gauge sourcecurrents, for instance,

Insisting on chiral gauge current conservation we are formally back to therenormalization conditions (13.10) and the anomaly is placed in the vector current.Notice that the condition of vanishing divergence of two currents can always beimposed on triangle diagrams. Restrictions on the fermion representations appearwhen we consider triangle diagrams with three gauge currents.

Since U (1) current is an SU (2) singlet, in the operator notation we have,analogously to (13.35),

∂µ jµ = −(1/16π2)Tr[εµνρσ GµνGρσ ] (13.62)

where Gµν = igGaµνT a is the SU (2) field strength tensor. An interesting

possibility is that the vacuum expectation value of the r.h.s. of (13.62) does notvanish due to some non-perturbative effects like tunnelling, with instantons as thedominant field configurations, or monopoles. One would then expect ‘topological’baryon number non-conservation.


Anomaly-free models

We conclude with more general remarks. As we know the triangle diagramanomaly is given by a multiple of

Dabc = 12 Tr[T a{T b, T c}] (13.63)

where T a , T b and T c are generators of the group of transformations of currentsattached to the three vertices. We take all three currents to be gauge sourcecurrents. One can distinguish three cases of anomaly-free models (Georgi &Glashow 1972). The right-handed fermion loop and the left-handed fermion loopanomalies may cancel. This happens when fermions with either of the chiralitiestransform according to the same representation of the gauge group, i.e. the theoryis vector-like (Appendix E). If this is not the case, for an anomaly-free theory thequantity Dabc must vanish. Two cases may be distinguished here. It may be that,for all representations of the group, Dabc = 0. These are called ‘safe’ groups. Theyinclude SU (2), all orthogonal groups except SO(6) ≈ SU (4) and all symplecticgroups. The other case is when the group is not safe, like SU (N ) with N > 2 orSU (2)×U (1), but there exist some ‘safe’ representations for which Dabc vanishes.This then gives a limitation on the allowed fermion representations.

13.3 Anomalies and the path integral

Introduction

For a systematic discussion of chiral anomalies it is convenient to use thegenerating functional and the path integral technique. We consider a theory ofmassless fermions coupled to non-abelian gauge fields. For the purpose of thepresent discussion it is convenient to treat the gauge fields as classical backgroundfields. Depending on the physical problem at hand we may need to use vectorand axial-vector couplings or left- and right-handed couplings. In the first case thegauge-invariant lagrangian density reads†

L = �iγ µ(∂µ + AVµ + γ5 AA

µ)� = �iγ µ� − jαµV AVαµ − jαµA AA

αµ (13.64)

where

AV,Aµ = iAV,A

αµ T α

and the T αs are the generators of the gauge group in fermion space. Thetransformation properties of the fermion and the gauge fields under infinitesimal

† Since in this section fields Aµ are treated as background fields, the coupling constant g is included in thedefinition of Aµ.

13.3 Anomalies and the path integral 479

vector and axial-vector gauge transformations are

� ′ = (1 −�V)�

δ�V AVµ = ∂µ�V − [�V, AV

µ] = Dµ�V

δ�V AAµ = −[�V, AA

µ]

(13.65)

� ′ = (1 − γ5�A)�

δ�A AVµ = −[�A, AA

µ]

δ�A AAµ = ∂µ�A − [�A, AV

µ]

(13.66)

respectively, where as usual � = i�αT α. We see, in particular, that in anon-abelian theory a pure axial-vector interaction with Dirac fermions wouldnot be gauge-invariant. The lagrangian (13.64) is also invariant under globalU (1) and UA(1) transformations. However, no gauge bosons are coupled to thecorresponding abelian currents.

It is often more convenient to use the left- and right-handed fields and then

L = �Liγ µ(∂µ + iAαLµT α)�L + �Riγ µ(∂µ + iAα

RµT α)�R

=∑L,R

�L,Riγ µ∂µ�L,R − jLαµ Aµ

Lα − jRαµ Aµ

Rα

(13.67)

where

jαL,Rµ = �L,RγµT αL,R�L,R

Invariance under gauge transformations on the chiral fields requires standardtransformation properties for the gauge fields Aµ

L,R (see Section 1.3). At theclassical level the U (1) currents are conserved: ∂µ jV

µ = ∂µ jAµ = 0 and the

non-abelian currents are covariantly conserved: Dµ jαµL,R = 0 (see (1.125)).Depending on the physical problem considered, the gauge fields in the la-

grangian (13.64) or (13.67) can be regarded as dynamical gauge fields or asauxiliary background fields, introduced to write down the generating functionalfor the anomalous Ward identities involving currents which in the framework ofthe considered theory are not coupled to any dynamical gauge fields. This is,for instance, the case with the flavour symmetry currents in QCD. Of course, thephysical interpretation of the non-abelian structure in the lagrangians (13.64) and(13.67) also depends on the problem. In addition the same fermions may couple toother gauge fields, too, corresponding to a gauge group acting in a different space.

Since, by assumption, the gauge fields are classical background fields, only twotypes of Feynman diagram appear in our theory: diagrams in which the externalfields couple to a free spinor line and diagrams in which the external fields couple


to a spinor loop. This, however, has no impact on the generality of the discussionof chiral anomalies due to the Adler–Bardeen non-renormalization theorem quotedin Section 13.1.

Abelian anomaly

Let us first consider the generating functional for Ward identities involving theU (1) axial-vector current. We use the lagrangian (13.64) and specify AA

µ = 0.Thus we consider the axial abelian anomaly in a QCD-like theory. Applying thegeneral formalism developed in Section 10.1 to derive Ward identities we take thefunctional

exp{iZ [Aµ, η, η]} =∫

D� D� exp

[i∫

d4x (Lkin− jµα Aαµ+η�+�η)

](13.68)

and use its independence of the choice of the integration variables changedaccording to the transformation

�(x) → exp[−iγ5�(x)]�(x), �(x) → �(x) exp[−iγ5�(x)] (13.69)

(the gauge fields Aαµ, are not altered). Naively we then get (10.9) which generates

Ward identities corresponding to ∂µ j5µ = 0 and is wrong in view of our perturbative

calculations and of (13.57) in particular. Using the latter we can anticipate thecorrect modification of (10.9). It reads

0 = δZ [A, η, η]/δ�|�=0

= 1

exp{iZ [A, η, η]}∫

d4z∫

D� D�

× exp

[i∫

d4x (�i/∂� − jµα Aαµ + η� + �η)

]×{−(n/8π2)Tr[Gµν(z)G

µν(z)] − ∂µ j5µ(z)+ η(−iγ5�)+ (−i�γ5)η}

(13.70)

One may ask where the anomalous term comes from in the path integral formalism.The answer has been given by Fujikawa (1980) who noticed that the anomalyis due to the non-invariance of the fermionic path integral measure under thetransformation (13.69). We shall come back to this point in Section 13.4.

Eq. (13.70) generates the desired Ward identities when we differentiate itfunctionally with respect to the Aα

µ, η and η and set η = η = 0. For instance

the anomalous Ward identity for the matrix element 〈0|T j5µ(z) jαν (y) jβδ (x)|0〉 is

obtained from (13.70) by taking the derivative

∂

∂ Aαν (y)

∂

∂ Aβ

δ (x)


and setting η = η = 0. One gets

∂µ〈0|T j5µ(z) jαν (y) jβδ (x)|0〉 = −(n/8π2)〈0|T Tr[GµρGµρ] jαν jβδ |0〉

and in momentum space this is just equivalent to (13.56). Differentiating withrespect to η and η and setting η = η = 0 one gets the anomalous Ward identity forthe matrix element 〈0|T� j5

µ�|0〉.

Non-abelian anomaly and gauge invariance

Our next task is to study the generating functional for the anomalous Wardidentities involving non-abelian currents. We again consider the functional

exp{iZ [Aµ]} =∫

D� D� exp

(i∫

d4x �iD/�

)(13.71)

corresponding to the lagrangian (13.64) or (13.67). The � stands for Dirac orchiral fermion fields and Aµ for vector and axial-vector gauge fields or left- andright-handed gauge fields, respectively; other fields are suppressed. First, westudy the gauge dependence of the functional Z [Aµ]. The gauge transformationof the fermion fields is again just a change of integration variables but under theaccompanying transformation of the gauge fields Aµ(x) → Aµ(x) + δAµ(x) thefunctional Z [Aµ] changes as follows

Z [Aµ] → Z [Aµ] − 2∫

d4x Tr

[δZ [Aµ]

δAµ(x)δAµ(x)

](13.72)

whereδ

δAµ

= δ

δAαµ

(iT α)

and

δZ [Aµ]

δAµ(x)= 1

exp{iZ [Aµ]}∫

D� D� jµ(x) exp

(i∫

d4x �iD/�

)≡ Jµ (13.73)

and jµ(x) is the current coupled to the field Aµ(x) whereas Jµ(x) is defined by(13.73). The explicit formulae (13.71)–(13.73) and that for δAµ depend, of course,on whether we use vector and axial-vector fields or chiral fields. If we work withvector and axial-vector gauge fields, then Aµ(x) in (13.72) includes both AV

µ andAA

µ which transform under vector or axial-vector gauge transformation accordingto (13.65) and (13.66). On the other hand, if we use chiral fields, the gaugetransformations on the left- and right-handed fields separately form a group andwe have

δ� AL,Rµ = ∂µ�

L,R − [�L,R, AL,Rµ ] = Dµ�

L,R (13.74)


In each case we may rewrite (13.72) and (13.73) using the following relation∫d4x Tr[ Jµ(x)Dµ�(x)] = −

∫d4x Tr[Dµ Jµ(x)�(x)] (13.75)

which is easy to verify. With the help of (13.75) we get

δ�Z = 2∫

d4x Tr[X (x)Z [Aµ]�(x)] (13.76)

or

δ�Z [Aµ]/δ�α(x)|�(x)=0 = −Xα(x)Z [Aµ] (13.77)

where

XV(x) = ∂µδ

δAVµ(x)

+[

AVµ(x),

δ

δAVµ(x)

]+[

AAµ(x),

δ

δAAµ(x)

](13.78)

(where X = iXαT α) for the vector gauge transformations on the vector and axial-vector fields,

XA(x) = ∂µδ

δAAµ(x)

+[

AVµ(x),

δ

δAAµ(x)

]+[

AAµ(x),

δ

δAVµ(x)

](13.79)

for the axial-vector gauge transformations on the vector and axial-vector fields, and

XL,R(x) = ∂µδ

δAL,Rµ (x)

+[

AL,Rµ (x),

δ

δAL,Rµ (x)

]= Dµ

δ

δAL,Rµ (x)

(13.80)

for chiral gauge transformations on the chiral gauge fields (all commutators are formatrices T α and not for fields which are c-numbers). By explicit calculation† oneverifies that the Xs satisfy the gauge group commutation relations

[XαV(x), Xβ

V(y)] = cαβγ Xγ

Vδ(x − y)

[XαV(x), Xβ

A(y)] = cαβγ Xγ

Aδ(x − y)

[XαA(x), Xβ

A(Y )] = cαβγ Xγ

Vδ(x − y)

[XαL,R(x), Xβ

L,R(Y )] = cαβγ Xγ

L,Rδ(x − y)

(13.81)

which, applied to Z [Aµ], are equivalent to the relation

δ�1δ�2 Z [Aµ] − δ�2δ�1 Z [Aµ] = δ[�1,�2] Z [Aµ] (13.82)

expressing the group property of the gauge transformations.

† It is convenient to use a test functional.


Eqs. (13.76), (13.77) and (13.81) or (13.82) have several implications. Firstly,for invariance of our quantum theory under one of the previously defined gaugetransformations we must have

δ�Z [Aµ] = 0 ⇒ X (x)Z [Aµ] = 0 (13.83)

where the X is one of the Xs given by (13.78)–(13.80) depending on the consideredgauge transformation. In a theory with only vector or only chiral gauge fields, using(13.73) we recover the equation

Dµ jVµ = 0 or Dµ jL,R

µ = 0 (13.84)

obtained in Section 1.3 on the classical level. The non-abelian source currentsfor gauge fields must be covariantly conserved at any order of perturbation theoryfor gauge invariance to be maintained in quantum theory. With dynamical gaugefields the generating functional Z [Aµ] differs from the Z [J, η, η] of the completetheory, where J, η, η are the gauge and fermion field sources, by the absence ofthe functional integration over DAµ and of the gauge field kinetic terms which aregauge-invariant. To study gauge invariance of the complete quantum theory it issufficient to study the gauge invariance of the Z [Aµ].

The anomaly means that the variation of Z [Aµ] under gauge transformation doesnot vanish

Dµ jL,Rµ (x) = GL,R(x) (13.85)

or equivalently

−XαL,R(x)Z [Aµ] = Gα

L,R(x) (13.86)

Similarly, if we work with vector and axial-vector currents and insist on the vectorcurrent conservation Dµ jV

µ (x) = 0 then the divergence of the axial current isanomalous and

Dµ jAµ = ∂µ jA

µ + [Aµ

V, jAµ ] = GA(x) (13.87)

or

−XαA(x)Z [Aµ] = Gα

A(x) (13.88)

Indeed, from the triangle diagram calculation in Section 13.1 we know partialcontributions to the GL,R

α (x) in (13.86) and the GAα (x) in (13.88) which are (at

the triangle diagram level Dµ = ∂µ)

±(1/12π2)εµνρσ Tr[T α∂µ AL,Rν ∂ρ AL,R

σ ] (13.89)

and

−(1/2π2)εµνρσ Tr[T α(∂µ AVν ∂ρ AV

σ + 13∂µ AA

ν ∂ρ AAσ )] (13.90)


respectively. In the second expression the first term comes from the AVV diagramand the second one from the AAA diagram. Different numerical coefficients reflectdifferent renormalization conditions for the two diagrams: conservation of thevector currents for the first one and symmetry between all three vertices for thesecond one. Observe that as long as for the anomaly the minimal form dependingonly upon the gauge fields and not other external fields is taken, (13.86) and (13.88)follow from (13.85) and (13.87), respectively, since the Gα(A)s factorize out of thepath integral which cancels with the exp{−iZ [A]}.

One way to get the full anomalous terms Gα(A) is to use the so-called Wess–Zumino consistency condition (Wess & Zumino 1971). Using (13.81) or (13.82)and (13.85) we get for the anomaly GL,R

α (x) the equation

Xβ

L,R(x)GαL,R(y)− Xα

L,R(y)Gβ

L,R(x) = −cαβγ Gγ

L,R(x)δ(x − y) (13.91)

The importance of this condition follows from the fact that the operator X is non-linear in the gauge potential Aµ. Therefore the Wess–Zumino condition completelydetermines Gα[Aµ] once the term (13.89) is given. After some calculation we canget

GαL,R(x) = ±(1/12π2)Tr[T α∂µεµνρσ (Aν

L,R∂ρ Aσ

L,R + 12 Aν

L,R Aρ

L,R AσL,R)] (13.92)

The expression for the axial anomaly must satisfy the following consistencyconditions (again, see (13.81))

XVα GA

β (y) = cαβγ GAγ δ(x − y)

XAα GA

β (y)− XAβ GA

α (y) = 0

}(13.93)

and it is much more complicated (Bardeen 1969). All the anomalous Wardidentities follow from the relations (13.86) or (13.88) (we may need to introduceadditional sources to the functional Z ).

Consistent and covariant anomaly

Finally we study the transformation properties of the anomalous current. Naively,this current would be expected to transform covariantly under gauge transforma-tions. To determine the effect of the anomalies, in addition to the gauge variation

δ� Aµ = Dµ�

δ�Z [A] = −2∫

d4x Tr[(δZ/δAµ)δ� Aµ(x)] = 2∫

d4x Tr[Dµ Jµ�(x)]

= −∫

d4x �α(x)Gα(A)

(13.94)


we introduce the variation which defines the current. By definition

δB Z [A] = −2∫

d4x Tr[(δZ/δAµ)δAµ] = −2∫

d4x Tr[ JµBµ(x)] (13.95)

and

δB Aµ(x) = δAµ ≡ Bµ(x)

The following operator equation holds

δBδ� − δ�δB = δ[B,�] (13.96)

Applying this operator to the functional Z [A] we obtain

(δBδ� − δ�δB)Z [A] =∫

d4x {[δB Gα(A)]�α(x)− (δ� Jµα )Bα

µ(x)}

=∫

d4x Jµα {[Bµ(x),�(x)]}α

(13.97)

The gauge transformation properties of the non-abelian current follow from (13.97)∫d4x (δ� Jµ

α )Bαµ(x) = −

∫d4x {[�, Jµ]α Bα

µ − [δB Gα(A)]�α} (13.98)

The first term on the r.h.s. gives the usual transformation property of the current;the second term is due to the anomaly. One can show (Bardeen & Zumino 1984)that in the case of the anomaly (13.92) there exists the covariant non-abelian current

J L,Rµα = J L,Rµ

α + PL,Rµα (A) (13.99)

Its explicit form can be found by searching for a polynomial PL,Rµα in gauge fields

with an anomalous gauge transformation rule opposite to that of the current Jµα .

Bardeen & Zumino (1984) show that

PL,Rµα = ∓(1/24π2)εµνρσ Tr[Tα(AνGρσ + Gρσ Aν − Aν Aρ Aσ )] (13.100)

The current J αµ is also anomalous

(Dµ J L,Rµ)α = Gα(A) = ±(1/16π2)εµνρσ Tr[TαGµνGρσ ] (13.101)

The anomalies (13.92) and (13.101) are called consistent and covariant anomalies,respectively. Each can be used to study anomaly cancellation. The physical senseof the original non-abelian current has been discussed earlier. The covariant currentmay have some physical significance when coupled to other external non-gaugedfields.


13.4 Anomalies from the path integral in Euclidean space

Introduction

In this section we discuss the techniques of anomaly calculation using the pathintegrals in Euclidean space (Fujikawa 1980). First let us summarize the rules ofcontinuation from Minkowski to Euclidean space. For the spatial coordinates wehave (see Section 2.3)

x0 = −ix4 xi = (−x,−y,−z)

xi = xi xµ = −xµ

}(13.102)

where µ = 1, 2, 3, 4 and correspondingly

p0 = −i p4 pi = (−px ,−py,−pz)

pi = pi pµ = i(∂/∂ xµ)

}(13.103)

where again µ = 1, 2, 3, 4. We define the Euclidean vector potential Aµ as

A0 = i A4, Ai = − Ai (13.104)

so that Aµ form a Euclidean four-vector. Thus for the covariant derivative Dµ =∂µ + Aµ = ∂µ + ig Aa

µT a we get (∂µ = (∂/∂x0,−∂/∂xi ))

D0 = iD4, Di = −Di , Dµ = −∂/∂ xµ + ig AaµT a (13.105)

The field strength tensor is

Gai j = Ga

i j (i, j = 1, 2, 3), Ga0i = −iGa

4i (13.106)

where

Gaµν = − ∂

∂ xµAa

ν +∂

∂ xνAa

µ + gcabc Abµ Ac

ν

We define the Euclidean γ -matrices to be hermitean matrices γn obeying

{γµ, γν} = 2δµν µ, ν = 1, . . . , 4 (13.107)

and explicitly

γ4 = γ0, γi = −iγi , i = 1, 2, 3 (13.108)

The matrix γ5 is taken as

γ5 = −γ1γ2γ3γ4 (13.109)

and it is also hermitean. The Dirac operator iD/ is hermitean. The fermion fields� and � are independent variables in the path integral and in Euclidean space wedefine

� = �, ˆ� = i� (13.110)

13.4 Anomalies from the path integral in Euclidean space 487

Under infinitesimal rotations in Euclidean space � and �† transform as follows:

δ� = 18(γµγν − γν γµ)ωµν� (13.111)

where

ωi j = ωi j , ω0i = iω4i

and

δ�† = − 18�

†(γµγν − γν γµ)ωµν (13.112)

so that �†1 �2 is a scalar and we identify the transformation properties of ˆ� as those

of �† (in Minkowski space �†1γ0�2 is a scalar). The Euclidean action is

iS = −S (13.113)

where

S =∫

d4 x[

14 Ga

µνGaµν + ˆ�(iD/ + iM)�

]To complete this introduction let us discuss the eigenvalue problem for the

hermitean Dirac operator iD/ in Euclidean space. For simplicity we assume that iD/has a purely discrete spectrum. This corresponds to stereographically projectingEuclidean four-space onto a four-sphere and changes the determinant but only bya factor which is independent of the gauge field. Since iD/ is hermitean it has realeigenvalues λn

iD/ �n = λn�n (13.114)

and because

{γ5, iD/ } = 0 (13.115)

we get

iD/ γ5�n = −λn γ5�n (13.116)

Thus non-vanishing eigenvalues always occur in pairs of opposite sign. We observealso that for λn �= 0 chiral fields are not eigenvectors of iD/

iD/ (1 ± γ5)�n = λn(1 ∓ γ5)�n (13.117)

On the other hand eigenfunctions of the vanishing eigenvalue can always be chosento be eigenfunctions of γ5 and thus to be chiral fields. Since γ 2

5 = 1 its eigenvaluesare ±1 and

γ5(1 ± γ5)�n = ±(1 ± γ5)�n (13.118)

In the rest of this section we work in Euclidean space and the caret is omitted.


Abelian anomaly with Dirac fermions

As mentioned in the previous section the anomaly of the ungauged axial U (1)current j5

µ = �γ µγ5� can be understood as a non-invariance of the path integralmeasure under local transformations on fermion fields

� ′ = exp[i�(x)γ5]� ≈ (1 + i�(x)γ5)�

� ′ = � exp[i�(x)γ5] ≈ �(1 + i�(x)γ5)

}(13.119)

The generating functional in the Euclidean space

exp{−Z [A]} =∫

D� D� exp

(−∫

d4x �iD/�

)(13.120)

where � and � are two independent variables describing Dirac fermions coupledto gauge potentials transforming under a vector-like gauge group, can be definedprecisely by expanding � and � in terms of the eigenfunctions ϕn of iD/

�(x) =∑

n

anϕn(x), �(x) =∑

n

ϕ†n(x)bn

iD/ ϕn(x) = λnϕn(x),∫

d4x ϕ†n(x)ϕm(x) = δmn

(13.121)

The coefficients an and bn are Grassmann variables. The measure becomes

D� D� =∏

n

dan

∏m

dbm (13.122)

up to an irrelevant arbitrary normalization factor. Under local infinitesimal axialtransformations (13.119) the lagrangian is transformed into

L′(x) = L(x)+ ∂µ�jµ5 (x)

and therefore∫d4x �iD/� →

∫d4x �iD/� −

∫d4x �(x)∂µ jµ5 (x) (13.123)

If the measure were invariant under axial transformations then using the invarianceof the path integral under a change of variables one could derive (10.11) (withδaL(x) = 0) implying the conservation of axial current at the quantum level.However, the measure (13.122) is not invariant and following Fujikawa we canexplicitly evaluate the Jacobian of the change of variables. We get

� ′(x) =∑

n

a′nϕn(x) =∑

n

an exp[i�(x)γ5]ϕn(x) (13.124)

or

a′m =∑

n

∫d4x ϕ†

m(x) exp[i�(x)γ5]ϕn(x)an =∑

n

Cmnan (13.125)


Thus (see (2.116)) ∏m

da′m = det−1 C∏

n

dan (13.126)

and the Jacobian for D� gives the identical factor. Furthermore, we have

C = exp[i�(x)γ5]

and

det C = exp{i Tr[�(x)γ5]} = exp

[i∑

n

∫d4x �(x)ϕ†

n(x)γ5ϕn(x)

](13.127)

where the trace is taken over the whole Hilbert space. Hence the measure ismultiplied by

exp

[−2i

∫d4x �(x)

∑n

ϕ†n(x)γ5ϕn(x)

](13.128)

which is ill defined as it stands and must be regularized. In our problem we want tochoose a regularization procedure which preserves the invariance under the vector-like gauge group with gauge potentials Aµ. Actually, with this fact in mind we havebeen working in the basis (13.121) of the eigenfunctions of iD/ = i(/∂ + A/) whichare gauge-invariant, i.e. gauge transformed eigenfunctions are eigenfunctions ofthe gauge transformed operator. Hence we may simply regularize the expression(13.128) with a gaussian cut-off for large eigenvalues λn∫

d4x �(x)∑

n

ϕ†n(x)γ5ϕn(x)

≡ limM→∞

∫d4x �(x)

∑n

ϕ†n(x)γ5ϕn(x) exp(−λ2

n/M2)

= limM→∞

∫d4x �(x)

∑n

Tr[γ5 exp[−(iD/ )2/M2]ϕn(x)ϕ†n(x)]

= limM→∞

∫d4x �(x) lim

y→x{γ5 exp[−(iD/ x)

2/M2]}abkl

∑n

[ϕn(x)]bl [(ϕ∗n(y)]a

k

(13.129)

where Tr means the trace over the group and the Dirac indices. Going over to a


plane wave basis and using the completeness relation for the ϕn(x) we get∑n

[ϕn(x)]bl [ϕ∗n(y)]a

k

= δabδkl∑

n

〈x |n〉〈n|y〉 = δabδkl〈x |y〉

= δabδkl∫

d4k

(2π)4〈x |k〉〈k|y〉

= δabδkl∫

d4k

(2π)4exp(−ikx) exp(+iky) (13.130)

and therefore†∫d4x �(x)

∑n

ϕ†n(x)γ5ϕn(x)

= limM→∞

∫d4x �(x)

∫d4k

(2π)4Tr[γ5 exp(+ikx) exp[−(iD/ )2/M2] exp(−ikx)]

= limM→∞

∫d4x �(x) Tr[γ5([γ

µ, γ ν]Gµν)2

(1

4M2

)2 1

2!

∫d4k

(2π)4exp(+kµkµ/M2)

= 1

16π2

∫d4x �(x) Tr[GµνGµν] (13.131)

where the last trace is over the group indices only. Using (13.123), (13.128) andinvariance of the path integral under the change of variables we finally get

∂µ j5µ(x) = (i/8π2)Tr[GµνGµν] (13.132)

which corresponds to the previous result (13.57) obtained in Minkowski space bythe triangle diagram calculation.

As a by-product of this derivation of the abelian anomaly we can easilyunderstand the content of the so-called index theorem for the Dirac operator iD/ .The theorem relates the number of positive chirality minus the number of negativechirality zero modes of the operator iD/ to the topological charge (Pontryagin index)of the background gauge field Aµ. Using the result (13.131) for �(x) independentof x , observing that ∫

d4x∑

n

ϕ†n(x)γ5ϕn(x) (13.133)

vanishes for eigenfunctions with eigenvalue λn �= 0 since γ5ϕn has eigenvalue −λn

and is thus orthogonal to ϕn , and writing

ϕ†n(x)γ5ϕn(x) = 1

4ϕ†n(1 + γ5)

2ϕn − 14ϕ

†n(1 − γ5)

2ϕn

† In Euclidean space we define ε1234 = −1 and then Tr[γ5γµγ νγ ργ δ] = 4εµνρσ .


for the eigenfunctions with eigenvalue λn = 0, we get

n+ − n− = Q (13.134)

where n+ (n−) is the number of positive chirality (negative chirality) zero modes ofthe operator iD/ and Q is the topological charge defined by (8.62). As in Section 8.3we assume here that the background gauge fields rapidly approach the pure gaugeconfiguration of |xµ| → ∞ so that the Euclidean action is finite and the topologicalcharge Q is well defined. The topological charge density is connected to theanomaly density.

Non-abelian anomaly and chiral fermions

We consider now the structure of the non-abelian anomaly in theories of chiralfermions coupled to an external dynamical or auxiliary gauge field Aµ anddescribed by the effective action

exp{−Z [A]} =∫

D�LD�L exp

(−∫

d4x �LiD/�L

)(13.135)

The left-handed fermions �L and the right-handed fermions �L are independentvariables. In Section 13.3 we have understood the non-abelian anomaly as abreakdown of the gauge invariance of the functional Z [A]. This suggests a possibleapproach to its calculation using the path integral formalism: calculate

Z [A] = − ln det iD/ (A) (13.136)

and its change under gauge transformation. However, as we know from theintroduction to this section the operator iD/ (A) maps positive chirality spinors tonegative chirality spinors

iD/ (A)�L → �R

and consequently it does not have a well-defined eigenvalue problem and a well-defined determinant on the space of chiral fields. The only exception is whenchiral fermions transform as a real representation of the gauge group, i.e. left-and right-handed chiralities transform under the same representation. Then we canformulate our theory in terms of the Dirac fermions � = �L+�R and in Euclideanspace we are back to the hermitean eigenvalue problem iD/�n = λn�n with realλn . In this case det iD/ is real and, as we know from the previous discussion, it canalways be regularized in a gauge-invariant anomaly-free manner (in other wordspositive and negative chirality anomalies cancel each other).


Now consider chiral fermions in a complex representation R. We can alwaysimagine an extended theory with additional fermions in R∗ so that

Z R+R∗[A] = Z R[A] + Z R∗[A] = Z R[A] + Z∗R[A] = 2 Re Z R[A] (13.137)

But R+ R∗ is a real representation and consequently Z R+R∗[A] is gauge-invariant.We arrive at an interesting observation (Alvarez-Gaume & Witten 1983): inEuclidean space only the imaginary part of the functional Z R[A] may suffer fromthe anomalies which give its anomalous variation under gauge transformation.Equivalently, if det iD/ was defined in the space of chiral fermions, we could saythat only the phase of iD/ may be gauge-non-invariant. One can circumvent theproblem of defining det iD/ observing that it suffices to calculate the change ofln det iD/ under a gauge transformation. And

ln det iD/ (Ag)− ln det iD/ (A) = ln det[−iD/−1(A)iD/ (Ag)] (13.138)

is meaningful since D/−1 D/ maps a space of fermion fields with given chiralityinto itself. The overall phase of the determinant is irrelevant for us and so isthe dependence on A; we can put, for example, A = 0. Thus we see that thecalculation of the non-abelian anomaly from the path integral is a well-definedproblem. The main difficulty is to invent an appropriate regularization procedure.For actual calculation, and different regularization methods we refer the readerto the original literature (Fujikawa 1984, Balachandran, Marmo, Nair & Trahern1982, Alvarez-Gaume & Ginsparg 1984, Leutwyler 1984). Of course, the resultagrees with (13.93) which in Euclidean space reads

GαL = Dµ

∂Z

∂ AαµL

= − i

12π2εµνρσ Tr[T α∂µ(Aν∂ρ Aσ − 1

2 iAν Aρ Aσ )] (13.139)

Problems

13.1 Derive the anomalous Ward identity (13.21) starting with the unregularized integralrepresentation (13.5), performing a shift of the integration variable such that (13.3)is satisfied for this integral representation and studying the surface term induced bythis change of the integration variable in the linearly divergent integral.

13.2 Show that Ward identities for the Green’s function of the three-vector currents areanomaly-free. Discuss this point also in Euclidean space using the material ofSection 13.4.

13.3 In Fig. 13.6 replace photons by gluons and check that in QCD there is no anomalyfor the axial isospin current.

Problems 493

13.4 Check that

GαA = −(1/4π2)εµνστ Tr[λα[ 1

4 FVµν FV

στ + 112 FA

µν FAστ

+ 23 (Aµ Aν FV

στ − AµFVνσ Aτ + FV

µν Aσ Aτ )− 83 Aµ Aν Aσ Aτ ]]

where

Vµ = +iλi V iµ, Aµ = +iλi Ai

µ

FµνV = ∂µV ν − ∂νV µ + [V µ, V ν] + [Aµ, Aν]

FµνA = ∂µ Aν − ∂ν Aµ + [V µ, V ν] + [Aµ, Aν]

satisfies the consistency conditions (13.93).

13.5 The determinant of an operator A with real positive discrete and increasing withoutbound eigenvalues λn : A fn = λn fn can be defined as

det A =∏

n

λn = exp[−(dζA/dt)(0)]

where the so-called ζ -function, defined as

ζA(t) =∑

n

1/λtn

converges for Re t > 2 and can be analytically extended to a meromorphic functionof t with poles only at t = 1 and t = 2 and is regular at t = 0. Using the integralrepresentation for the �-function

�(t) =∫ ∞

0xt−1 exp(−x) dx

check that the ζA-function can be calculated as

ζA(t) = 1

�(t)

∫ ∞

0dτ τ t−1

∫dx h(x, x, τ )

where

h(x, y, τ ) =∑

n

exp(−λnτ) fn(x) f ∗n (y) = 〈x | exp(−Aτ)|y〉

and h(x, y, τ ) obeys the so-called ‘heat equation’

Ax h(x, y, τ ) = −(∂/∂τ)h(x, y, τ ), h(x, y, τ = 0) = δ(x − y)

Note also that

ζA(t, x, y) ≡∑

n

fn(x) f ∗n (y)

λtn

= 1

�(t)

∫ ∞

0dτ τ t−1 h(x, y, τ )

The last equation can be used to regularize quantities such as∑n

ϕ+n (x)γ5ϕn(x) = limt→0y→x

Tr[γ5ζA(x, y, t)]


with A = (D/ )2. The heat kernel h(x, y, τ ) then has the asymptotic expansion(Nielsen, Grisaru, Romer & van Nieuwenhuizen 1978)

h(x, y, τ ) = 1

16π2τ 2exp

[− (x − y)2

4τ

] ∞∑n=0

an(x, y)τ n

for small τ . Check that

ζ ˆ�D2(x, x, 0) = (1/16π2)a2(x, x)

13.6 Determine the abelian and non-abelian chiral anomalies in 2n-dimensional space-time (Frampton & Kephart 1983, Zumino, Wu & Zee 1984, Leutwyler 1984).

13.7 Prove (13.134) starting with the anomalous divergence in Euclidean space

∂µ j5µ = 2m�γ5� − 2iQ(x)

(Integrate over the Euclidean volume, take the vacuum expectation value and thentake the limit m → 0.)

14

Effective lagrangians

14.1 Non-linear realization of the symmetry group

Non-linear σ -model

To become familiar with the effective lagrangian technique it is useful to follow theoriginal approach and to begin with the discussion of the SU (2)× SU (2)-invariantσ -model in which the chiral symmetry has been spontaneously broken: SUL(2)×SUR(2) → SUV(2). We consider the σ -model with bosons only. The quadrupletof real fields �0 = (π1π2π3σ0)

T transforming as (2, 2) under the chiral groupsplits then into massless Goldstone bosons and a massive σ = σ0 − v mesonwhich transform under the unbroken subgroup SU (2) as a triplet and a singlet,respectively. The σ mass is proportional to the vacuum expectation value v

and can be made arbitrarily heavy. Imagine we are interested in the physics atlow energy, E � mσ , only. It is natural to expect that the low energy regioncan be effectively described in terms of the Goldstone boson fields, with the σ

meson decoupled. However, an inspection of the lagrangian (9.38) shows thatthis does not happen: the ππσ coupling is proportional to v and therefore, forexample, the σ -exchange contribution to the low energy ππ → ππ scatteringcannot be neglected when mσ → ∞. The reason is that a naive decouplingof the σ field by its elimination from the lagrangian (9.38) explicitly breaks theSU (2) × SU (2) invariance of the lagrangian. Both the ππππ and the ππσ

couplings originate from the SU (2) × SU (2) invariant term (σ 20 + πππ2)2 in the

original lagrangian. Equivalently, decoupling of the σ means setting σ0 = v andthis breaks the invariance since (πππ, σ0) does no longer transform as a quadrupletunder SU (2) × SU (2). Can we find an effective low energy description in termsof the Goldstone bosons only which would be SU (2)× SU (2)-invariant?

Our goal is to implement the SU (2) × SU (2) symmetry in such a way thatthe two irreducible multiplets, the triplet and the singlet of the unbroken SU (2)which are contained in the quadruplet �0 do not mix under general SU (2)×SU (2)

495

496 14 Effective lagrangians

transformations. This can be achieved if we use the freedom of choice of the brokensymmetry vacuum (Section 9.5) among the degenerate set |α〉

|α〉 = exp(+iαa Xa)|0〉 (14.1)

The corresponding transformation of the field operators reads

�′ = exp(iαa Xa)� exp(−iαa Xa) (14.2)

or in the matrix form

�′ = exp(−iαa Xa)� (14.3)

where the Xas are broken generators in (14.1) and (14.2) and their matrixrepresentation in (14.3). Let us take some �0 = (πππ(x), σ0(x))T in the originallydefined vacuum. First of all we observe that at a given point x the Goldstone bosonfields can always be set to zero by a suitable redefinition of the vacuum. In factusing the real representation (E.38) for the generators we can write

�0(x) =

π1(x)

π2(x)

π3(x)

σ0(x)

= exp(−iξa Xa)

0

0

0

σ(x)

=

−σ(ξ1/ξ) sin ξ



σ cos ξ

(14.4)

where

ξ = (ξ 21 + ξ 2

2 + ξ 23 )

1/2

and

σ 2(x) = πππ2(x)+ σ 20 (x)

and therefore due to (14.3) a rotation of the vacuum by the angle ξξξ(x) sets the newGoldstone boson fields to zero locally at x . Any chiral transformation on the vector�0(x) can then be written as follows:

�′0(x) = exp(−iαaGa)�0(x) =

π ′

1(x)

π ′2(x)

π ′3(x)

σ ′0(x)

= exp[−iξ ′a(ξ, α)Xa]

000

σ(x)

(14.5)

where

Ga ≡ (Xa, Ya), Tr[G j Gi ] = δ j i

and Xa and Ya are broken and unbroken generators of SU (2)×SU (2), respectively,

14.1 Non-linear realization of the symmetry group 497

with the same σ(x) as in (14.4) because σ 2(x) = πππ2(x)+ σ 20 (x) is invariant under

chiral rotations. Thus, a general chiral transformation on �0(x) can be representedby the new angles ξ ′a(ξξξ,ααα) specifying the vacuum orientation which sets thetransformed Goldstone boson fields locally to zero. The vector (0 0 0 σ(x))T refersnow to the new, rotated, vacuum.

Since σ(x) is an SUV(2) singlet we suspect that in general a chiral transfor-mation can be factorized into an unbroken-SU (2)-subgroup transformation (whichin our case acts on the SU (2) singlet σ(x)) and a rotation of the local vacuumorientation. However, before we generalize our discussion, for the purpose offurther considerations let us check the transformation properties of the ξa(x)s underthe unbroken SU (2) isospin group

exp(−iuaYa)�0 = exp(−iuaYa) exp(−iξa Xa) exp(iuaYa) exp(−iuaYa)

000σ

= exp(−iξa Rab Xb)

000σ

= exp(−iξ ′b Xb)

000σ

(14.6)

Henceξ ′b = Rabξa

R is the matrix representation of the unbroken SU (2) under which the matrices Xa

transform

exp(−iucYc)Xa exp(iucYc) = Rab Xb

i.e. ξa(x)s indeed transform linearly under the SUV(2), belong to the samerepresentation as Xas and transform according to RT. These results can begeneralized as follows (Coleman, Wess & Zumino 1969, Callan, Coleman, Wess& Zumino 1969). We consider a theory with some symmetry group G whichis spontaneously broken to a subgroup H . Let G be a compact, connected,semi-simple Lie group. From the properties of the exponentials it follows that, insome neighbourhood of the identity of G, every group element can be decomposeduniquely into a product of the form

g = exp(iξa Xa) exp(iuaYa) (14.7)


where Xas and Yas are broken and unbroken generators of the group G, respec-tively.† If �0 is a field which transforms according to a linear representation of G,we can recast it as

�0(x) = exp[−iξa(x)Xa]�(x) ≡ U� (14.8)

(Xa is the matrix representation of the broken generator Xa appropriate for �0(x))where under a general G transformation g(α)

ξξξ ′ = ξξξ ′(ξξξ,ααα)�′ = exp[−iui (ξξξ,ααα)Yi ]�

}(14.9)

and ξξξ ′(ξξξ,ααα) and u(ξξξ,ααα) are non-linear functions of ξξξ and ααα. Indeed, using (14.7)

�′0(x) = exp(−iαaGa)�0(x) = exp(−iαaGa) exp[−iξa(x)Xa]�(x)

= exp[−iξ ′a(ξξξ,ααα)Xa] exp[−iua(ξξξ,ααα)Ya]�(x) (14.10)

since exp(−iα ·G) exp(−iζ ·X) belongs to G and can be decomposed as in (14.7).‡Thus, a general global transformation belonging to the group G is realized as anon-linear transformation on the fields ξa(x) and as a gauge (local) transformation,belonging to the unbroken H , on the multiplet �. We say that the fields (ξ,�)

provide a non-linear realization of the group G. Under transformations belongingto the unbroken subgroup H the fields ξi (x) transform linearly, according to thegeneralized (14.6).

Let us go back to the σ -model. Given the transformation rule (14.5) we observethat the field σ(x) can be decoupled at E � mσ in a chirally invariant manner bysetting σ(x) = v or equivalently by imposing on the field �0(x) a chirally invariantconstraint �T

0�0 = v2. After rescaling �0/v → �0 we obtain the so-called non-linear σ -model described by the lagrangian

L = 12 f 2

π ∂µ�T0∂

µ�0, �T0�0 = 1, f 2

π = v2 (14.11)

Imagine that using the lagrangian (14.11) we want to calculate matrix elementsinvolving Goldstone bosons. We can work either with Green’s functions of currentsor with the on-mass-shell matrix elements. The first method is more general (see,for example, Gasser & Leutwyler (1984)) and consists in gauging the full chiralgroup, introducing the external gauge fields by changing the ordinary derivativesacting on the linearly transforming fields �0 into covariant derivatives and expand-ing the generating functional, given by the vacuum-to-vacuum amplitude in the

† The scalar fields ξ(x) are coordinates of the manifold of left cosets G/H at each point of space-time. Theset of group elements l(ξ) = exp(iξa Xa) parametrizes this manifold. Eq. (14.7) means that once we havea parametrization l(ξ) each group element g can be uniquely decomposed into a product g = l · h, whereh ∈ H . The l is the representative member of the coset to which g belongs and h connects l to g within thecoset.

‡ A product of g with an arbitrary group element and in particular with some l(ξ) defines another l(ξ ′) and anh according to g · l(ξ) = l(ξ ′) · h, where ξ ′ = ξ ′(ξ, g) and h = h(ξ, g).


presence of external gauge fields, in powers of external fields. If we are interestedin the on-mass-shell matrix elements only, it is best to interpret the transformingnon-linearly ξa(x)s, rather than the πa(x)s, as the pion fields (Weinberg 1968,Coleman, Wess & Zumino 1969). Such a redefinition of the pion field does notchange the S-matrix elements if the redefinition is local and leaves the origin offield space invariant and as long as the new object has the right quantum numbers.Different definitions lead to different matrix elements off the mass-shell but theyall give the same results on the mass-shell. We shall briefly discuss this secondmethod. Irrespective of the method, however, one finds that beyond the tree levelour theory has UV divergences which require counterterms of higher and higherorder in derivatives of fields reflecting the non-renormalizability of the non-linearσ -model. This leads us to the following important comment on the physicalsignificance of the effective lagrangian.

We have arrived at the lagrangian (14.11) by freezing the σ degree of freedomin the linear σ -model. However, one can convince oneself that this lagrangian isthe most general one in order p2 which is Lorentz-, parity- and chiral-invariant andcan be constructed in terms of the real O(4) vector �0 of unit length �T

0�0 = 1.†Thus we can think about �0 as representing the degrees of freedom of compositeGoldstone bosons in the SU (2) × SU (2)-invariant QCD as well. The real virtueof the effective lagrangians is that they provide the most general systematic lowenergy expansion in the light particle sector, invariant under full symmetry Gof the underlying theory. In this context the non-renormalizability of (14.11) isnot a problem. Working to order p4 or higher we anyway have to supplementthe lagrangian (14.11) with additional terms with free parameters. Let us againconsider QCD. Its chiral symmetry is spontaneously broken into the isovectorsubgroup H . There are Goldstone bosons, composites of quark fields which form amultiplet under H , and possibly other states also with well-defined transformationproperties under H . Due to spontaneous symmetry breaking, chiral multipletssplit into H -multiplets with quite different masses in general. Imagine we areinterested in the low energy description of the strong interactions. We are notable yet to get it directly from QCD in terms of its fundamental parameter �

(and quark masses, if explicit breaking of chiral symmetry is taken into account).There is also no reason to expect that the linear σ -model is a good approximationto the low energy QCD. On the other hand the non-linear effective lagrangian isa tool for exploiting QCD as much as possible without solving the confinementproblem, namely for exploiting its underlying chiral symmetry. This happens atthe expense of a set of new unknown constants at every level of the low energyexpansion.

† �T0 ∂µ�0 vanishes and �T

0 ∂µ∂µ�0 = −∂µ�T

0 ∂µ�0.


Effective lagrangian in the ξa(x) basis

We return to the problem mentioned earlier of calculating the on-mass- shell matrixelements from effective lagrangians. Because of the constraint �T

0�0 = 1 the form(14.11) is not convenient for constructing the Feynman rules. They can be derivedeasily if we work in the ξa(x) basis. We can introduce the covariant derivative ofthe field ξ(x), defined by its transformation properties under general transformationG(α) as follows

[(Dµξ)a(x)]′ ≡ (Dµξ)b Rba(exp[−iuc(ξξξ,ααα)Yc]) (14.12)

where the matrix Rba is defined by the relation

exp[−iua(ξξξ,ααα)Ya]Xb exp[iua(ξξξ,ααα)Ya] = Rbc(exp[−iua(ξξξ,ααα)Ya])Xc (14.13)

and Ya and Xa are representation matrices of unbroken and broken generators of G,respectively, in the representation to which the �0 belongs. The functions ua(ξ, α)

are defined by (14.9). The covariant derivative can be constructed explicitly byconsidering the derivative ∂µ�0 which transforms under G transformations in thesame way as �0 does. Using the non-linear notation (14.8) for a vector �0 we can,on general grounds, expand as follows:

∂µ(U�) = exp[−iξa(x)Xa]{i∂µξa(x)Dab(ξ)Xb�+ [i∂µξa(x)Eab(ξ)Yb + ∂µ]�}(14.14)

and correspondingly for ∂µ(U�)′ with ξa → ξ ′a , Dab → D′ab, Eab → E ′

ab, � →�′, where the functions D(ξ) and E(ξ) are defined by†

exp(iξa Xa)∂µ exp(−iξa Xa) = i∂µξa[Dab(ξ)Xb + Eab(ξ)Yb] (14.15)

The transformation properties under G of both sides in (14.14) are the same asthose of the �0, so the object in the braces must be a multiplet of H and have thelocal gauge transformation law (14.9). Thus

[i∂µξ′a(ξξξ,ααα)D′

ab(ξ′)Xb + (i∂µξ

′a E ′

ab(ξ′)Yb + ∂µ)]�

′

= exp(−iucYc)[i∂µξa Dab Xb + (i∂µξa EabYb + ∂µ)] exp[iuc(ξξξ,ααα)Yc]�′︸︷︷︸�

Identifying

(Dµξ)b = i∂µξa Dab(ξ) ≡ Tr[U †∂µU Xb] (14.16)

Dµ� = [∂µ + i∂µξa Eab(ξ)Yb]� ≡ {∂µ + Tr[U †∂µUYb]Yb}� (14.17)

† Those who are familiar with differential geometry will recognize that in the expansion (14.15) Dab(ξ)represents the vielbein in G/H and Eab(ξ) is an H -connection.


and equating coefficients of Xa we recover for the object (Dµξ)b the transformationlaw (14.12). Similarly we also get

(Dµ�)′ = exp[−iua(ξξξ,ααα)Ya]Dµ� (14.18)

Thus, under general transformation G(α) functions �, Dµ� and (Dµξ)b alltransform according to some finite matrix representations of the group H . Animportant conclusion following from these transformation rules is that a functionof �, Dµ� and (Dµξ)b which is invariant under H transformations is alsoautomatically invariant under G transformations.

The explicit form of the covariant derivative can be obtained from (14.15). First,expanding the l.h.s. (14.15) in power series one can check the identity

exp(iξa Xa)∂µ exp(−iξa Xa)

=∞∑

n=0

(−i)n+1

(n + 1)![. . . [[∂µξa · Xa, ξb ·

1

Xb], ξd ·2

Xd] . . .. . .

ξw ·n

Xw] (14.19)

The commutators give

∂µξα · icabcξb1

· iccdeξd2

. . . icuwz. . .

ξwn

Gz

where

[Ga, Gb] = icabcGc

is the G group algebra and Gz is a broken generator X for n even, and unbroken Yfor n odd. Denoting

icabcξb = (c · ξ)ac (14.20)

one gets

exp(iξa · Xa)∂µ exp(−iξa · Xa) =∞∑

n=0

(−i)n+1

(n + 1)!∂µξa(c · ξ)n

azGz (14.21)

and summing the terms even in n

Dab(ξ) =∞∑

m=0

(−1)m+1

(2m + 1)!(c · ξ)2m

ab = −(

sin c · ξc · ξ

)ab

(14.22)

Similarly

Eab(ξ) =∞∑

m=0

i(−1)m

(2m + 2)!(c · ξ)2m+1

ab = i

(1 − cos c · ξ

c · ξ)

ab

(14.23)


Using the transformation properties of the covariant derivative (Dµξ)a theeffective lagrangian of the non-linear σ -model can be written down in the ξ(x)basis. It reads†

L = 12 f 2

π (Dµξ)a(Dµξ)a (14.24)

As expected, there is no dependence on the ξ field other than that present in thecovariant derivative. Any such dependence would break G invariance because onemay always set ξ(x) to zero at any given x by rotating the vacuum. Using theexplicit form of the covariant derivative we can now easily read off the Feynmanrules.

We can extend the effective lagrangian description to include the interaction ofGoldstone bosons with other fields which may be relevant for the effective lowenergy theory. According to the previously described general approach a multiplet�0 transforming linearly under G is replaced by the fields (ξ(x),�), where

�0 = exp[−iξa(x)Xa]�

providing a non-linear realization of the group G which is linear on the unbrokensubgroup H . Any H -invariant form built of �(x), Dµ�(x) and (Dµξ(x))a isautomatically G-invariant. For definiteness let us take � to be a fermion field andG to be the chiral symmetry spontaneously broken to the isospin subgroup H . Theleading low energy terms are

��, �γµγ5Y a�(Dµξ)a, �γµDµ�, . . .

The presence of the first term reflects the fact that the chiral symmetry of thelagrangian tells us nothing about the fermion masses as the symmetry may bespontaneously broken. The second term contains the fermion–fermion–pion vertexand the effective gradient coupling is known as the Adler zero for the soft pionemission. If we want to use the effective lagrangian beyond the tree approximation,terms of higher order in derivatives must be systematically included.

Matrix representation for Goldstone boson fields

In QCD-like theories with chiral U (N )×U (N ) symmetry spontaneously broken toSUV(N )×UV(1)‡ it is very convenient to collect the Goldstone boson fields into aunitary matrix U transforming linearly under chiral U (N )×U (N ) transformations.Let us consider a general matrix

� = �L j�Ri (14.25)

† Or L = 12 f 2

π gab∂µξa∂µξb , where gab is given in terms of the vielbein Dab(ξ) and is a G-invariant metricon G/H .

‡ See Section 13.2 for the discussion of the spontaneous breaking of UA(1).


where � iL,R are chiral fermion fields with flavour i . Under chiral transformations

� transforms linearly and according to (9.85)

�′ = UR�U †L (14.26)

Matrices UR and UL act in the space of fermions �R and �L, respectively, and havethe form

UL,R = exp(−iαaL,RT a) (14.27)

where a = 0, 1, . . . , N 2 − 1 and T 0 = 1l and T a , a = 1, 2, 3, . . . , N 2 − 1,are the SU (N ) generators in the fermion representation with Tr[T aT b] = 1

2δab.

The transformations under the unbroken vector group UV(N ) are represented byαL = αR and axial transformations by αL = −αR. Observe that now we alwayswork with SU (N ) generators T a and do not need to introduce explicitly the brokenand unbroken generators of the full chiral SU (N )× SU (N ). Thus a change of thevacuum orientation among the degenerate set corresponds to � → �′, where

�′ = exp(iαaTa)� exp(iαaTa) (14.28)

where a = 0, 1, . . . , N 2 − 1. Imagine now that we take � = 1l which accordingto (9.87) is an ansatz for the spontaneous breaking SU (N )× SU (N ) → SUV(N ).The matrix

� = exp(2iαaTa)1l ≡ U · 1l (14.29)

corresponds to a rotated vacuum and represents the degrees of freedom necessaryto describe the Goldstone boson sector. We check that any matrix � written as aproduct of a unitary and hermitean matrix � = U · h transforms under generalchiral transformations as follows:

�′ = URUhU †L = URUU †

LULhU †L = U ′h′ (14.30)

so U also transforms linearly and h undergoes only vector transformations. Thusonce � is written in this product form any full UV(N ) multiplet can be decoupledfrom the matrix h in the chirally invariant way. Since (14.29) is in the product formit provides us with the desired chirally invariant decoupling of all but Goldstonedegrees of freedom. Actually, the unitary matrix U (x) contains N 2 fields

U (x) = exp[ 13 iα0(x)] exp[2iαa(x)Ta] (14.31)

where a = 1, . . . , N 2 − 1 and T a are hermitean and traceless. For instance, in thecase of U (3)×U (3) it includes the η′ degree of freedom which, due to the abeliananomaly, is not a Goldstone boson (see Section 13.2). The single component fieldα0(x) is related to the determinant of U

det U = exp(iα0) (14.32)


Under a UA(1) axial transformation

U ′ = exp(2iβ)U (14.33)

Under an arbitrary chiral SU (N )× SU (N ) transformation

exp(− 13 iα0)U

′ = exp(−iβaRT a) exp(2iαcT c) exp(iβa

LT a)

= exp[2iα′c(βR, βL, α)Tc] (14.34)

where c = 1, . . . , N 2−1 so the fields (ααα, h) provide a non-linear realization of thischiral group. One can also easily check that αas transform linearly under vectortransformations UR = UL.

Because U is an unitary matrix UU † = 1, it is impossible to write a U (3) ×U (3)-invariant interaction without derivatives. In order p2 there are two U (3) ×U (3) invariants

I1 = Tr[∂µU∂µU †]

I2 = −∂µα0∂µα0 = {Tr[U †∂µU ]}2

(from (14.32) α0 = −i ln det U = −i Tr ln U ). On the basis of arguments which arenot discussed in this book such as the Okubo–Zweig–Iizuka rule and the large Nc

(number of colours) limit of QCD (Veneziano 1979, Witten 1979, Di Vecchia 1980)one expects the I1 term to be the dominant one. Since

U−1∂µU ≡ i∂µαa Dab(αc)Tb ≡ (Dµα)bTb (14.35)

where Tb form a group, the effective lagrangian can now be written in one of thefollowing equivalent ways

L = 14 f 2

π Tr[∂µU∂µU †]

= 14 f 2

π Tr[U−1∂µU (U−1∂µU )†]

= 12 f 2

π Tr[U−1∂µU Ta] Tr[(U−1∂µU )†Ta]

= 18 f 2

π (Dµα)b(D α)†b

(14.36)

Using the definition (14.35) and (14.30) one can easily check that for general chiraltransformations the ‘covariant derivative’ (Dµα)a transforms only under globalrotations of the unbroken SUV(N ). Thus if we want to include in the effectivelagrangian other multiplets of the unbroken H we can work with their ordinaryderivatives.

14.2 Effective lagrangians and anomalies

In QCD and in models of composite quarks and leptons one starts from afundamental lagrangian describing a non-abelian gauge theory with elementary


fermions. However, usually we are unable to solve the original theory and find thespectrum and the interaction of the composite states. The virtue of the effectivelagrangian approach is that it provides a systematic method of isolating thecomposite states that are relevant at low energy and of studying their interactionsin terms of a finite number of free effective parameters. The basic requirement inthe construction of an effective lagrangian is that it possesses the same symmetriesas the original theory. If this theory has anomalies the effective lagrangian mustalso satisfy the same anomalous transformation laws. Then the anomalous Wardidentities will be satisfied at least to the tree order.

Abelian anomaly

In QCD the axial current has the UA(1) anomaly (Section 13.2)

∂µ jµ5 = −2N (1/32π2)εµνρσ GµνGρσ ≡ 2Nq(x) (13.57)

where N is the number of flavours, determined by the response of the theory to theaxial UA(1) transformations on the fermion fields. The effective lagrangian (14.36)describing the interaction of Goldstone bosons is, however, invariant under theUA(1) transformations. We want to generalize our effective theory so that (13.57) issatisfied by virtue of the equations of motion. Then the anomalous Ward identitieswill be satisfied to the tree order. The problem can be solved (Rosenzweig,Schechter & Trahern 1980, Di Vecchia & Veneziano 1980, Witten 1980) byintroducing the pseudoscalar ‘glueball’ field q(x) into the effective lagrangianwhich is modified to

L = 14 f 2

π Tr[∂µU∂µU †] + 12 iq(x)Tr[ln U − ln U †] + cq2(x) (14.37)

Every term is invariant under SU (N ) × SU (N ) transformation. Under UA(1)transformation U ′ = exp(i�)U the action transforms as follows

S′ = S −�N∫

d4x q(x) (14.38)

The UA(1) matter current found by Noether’s theorem is

j5µ = 1

4 i f 2π Tr[∂µU †U −U †∂µU ] (14.39)

and therefore

∂µ j5µ = 1

4 i f 2π Tr[U�U † −U †�U ] (14.40)

On the other hand from the equations of motion we get

14 f 2

π (�U −U�U †U )+ iq(x)U = 0 (14.41)


Multiplying (14.41) on the left by U † and subtracting the complex conjugateequation gives

∂µ j5µ = 2Nq(x)

Once the effective field q(x) has been introduced into the lagrangian the presenceof the chirally invariant term q2(x) is crucial for the consistency of this approach†and for the successful phenomenology. The additional free parameter c is neededfor the phenomenological solution to the UA(1) problem, i.e. for the correctdescription of the masses of the pseudoscalar nonet (including η′) in the frameworkof the effective lagrangian approach (see Problem 14.1). Of course, to this endone must also introduce a term which accounts for the explicit breaking of thechiral symmetry by the quark masses. The quark mass terms transform underSU (3) × SU (3) as (3, 3) ⊕ (3, 3) so we must construct, from U , terms whichtransform under SU (3) × SU (3) in the same way. Since the components of Utransform as (3, 3), to lowest order in the number of derivatives the necessarycorrection is

L = 14 f 2

π {Tr[MU ] + Tr[M†U †]}

where M is some matrix which must be a multiple of the quark mass matrix.It is convenient to eliminate the field q(x) from the lagrangian (14.37) by using

its equation of motion. We finally get

L = 14 f 2

π Tr[∂µU∂µU †] + 14 f 2

π {Tr[MU ] + Tr[M†U †]} + 1

8c{Tr[ln U − ln U †]}2

(14.42)Effects connected to the existence of a non-vanishing vacuum angle � can also bestudied by adding to the lagrangian (14.42) a term �q(x).

The Wess–Zumino term

In Chapter 13 we related the presence of the non-abelian anomaly in theorieswith chiral fermions to the gauge invariance breaking of the functional Z [A], see(13.76):

δ�Z [A] =∫

d4x [−�a(x)Xa(x)]Z [A]

where A denotes dynamical or auxiliary gauge fields coupled to the fermioniccurrents and Z [A] is the effective action of the theory with fermions integrated

† In general arbitrary chiral-invariant functions of q(x) are possible. The term q2(x) is the only one whichsurvives in the limit Nc →∞, where Nc is the number of colours (see, for instance, Di Vecchia (1980)).


out

exp{iZ [A]} =∫

D� D� exp

(i∫

d4x L)

In Section 13.3 we have also found certain consistency conditions (integrabilityconditions) which must be satisfied in the presence of anomalies. Our questionnow is how to take account of the non-abelian anomaly using the effectivelagrangian for a QCD-like theory with SU (N )× SU (N ) symmetry spontaneouslybroken to SUV(N ). The Goldstone boson fields are collected in the matrixU = exp[2iαa(x)T a], a = 1, 2, . . . , N 2 − 1. The field α0(x) in (14.31) which is asinglet under SU (N ) × SU (N ) is irrelevant for the present discussion. The term(14.36) constructed in the previous section is chirally invariant and does not satisfythe Wess–Zumino consistency condition. To find the correct generalization of theeffective lagrangian we imagine an addition to the original fermionic lagrangian ofthe effective Goldstone fields coupled to fermions. When fermions are integratedout one now gets the effective action Z [αa, A] (rather than Z [A]) which shouldsatisfy the Wess–Zumino condition.

Let us consider a finite axial gauge transformation

Z [α′, A′] = exp

{∫d4x �a(x)[−XA

a (x)+ Va(x)]

}Z [α, A] (14.43)

where XAa (x) is given in (13.79) and Va(x) denotes an infinitesimal axial gauge

transformation operating on the Goldstone boson fields αa . Using the notation∫d4x �a(−XA

a + Va) = � · (−XA + V ) (14.44)

we observe that

[� ·(−XA+V )]n Z [α, A] = [� ·(−XA+V )]n−1(� ·GA) = (−� ·XA)n−1(� ·GA)

(14.45)where Ga

A is the axial anomaly, see (13.87), depending only on the gauge fields.Therefore the solution to (14.43) reads

Z [α′, A′] = Z [α, A] − exp(−� · XA)− 1

� · XA� · GA (14.46)

We know that for each ααα(x) there exists an axial rotation such that ααα′(x) = 0.Setting �a(x) = −αa(x) we get

Z [0, A′] = Z [α, A] − exp(α · XA)− 1

α · XAα · GA (14.47)

Since Z [0, A′] does not involve the fields αa(x) we can assume Z [0, A′] = 0 and


then

Z [α, A] = −1 − exp(α · XA)

α · XAα · GA =

∫ 1

0dt exp(tα · XA)α · GA (14.48)

The explicit solution for Z [α, A] can be used for reading off the vertices of theeffective lagrangian which describe the consequence of the anomalies in the Wardidentities. Expanding the exponential and using the explicit expression for XA,see (13.79), and GA (see Problem 13.4) one can work out the effective verticesorder-by-order in αa(x) = πa(x)/ fπ . An interesting point is that due to thederivatives with respect to the gauge fields A present in the XA the effective actiondoes not vanish for A = 0. Thus the anomalous effective lagrangian containsvertices describing interactions between Goldstone bosons. Using (13.79) andProblem 13.4 one finds, after a short calculation, that the lowest order term is

1

6π2 f 5π

εµνσρ Tr[π∂µπ∂νπ∂σπ∂ρπ ] (14.49)

where π = πaT a . In a similar way one can also work out vertices involvingpseudoscalars and one vector abelian gauge field describing the photon coupledto the neutral component of the SUV(N ) current (see Wess & Zumino (1971) andProblem 14.2).

Further attention has been paid to the construction and analysis of the Wess–Zumino terms in the effective action (see, for instance, Witten (1983), Adkins,Nappi & Witten (1983), Alvarez-Gaume & Ginsparg (1984)).

Problems

14.1 In terms of the free parameters M and c obtain the mass spectrum of the pseudoscalarmesons from the quadratic in the fields αa(x) part of the lagrangian (14.42); U =exp(iαi T i ), i = 0, 1, . . . , 8 (Veneziano 1979).

14.2 Using (14.48) and the axial anomaly GA(A) given in Problem 13.4 find the effectivevertices describing the processes π0 → 2γ , γ → π+π−π0 and γ + γ → 3π(consider the two-flavour case).

15

Introduction to supersymmetry

15.1 Introduction

This last chapter is intended as a brief introduction to supersymmetric fieldtheories. In recent years they have become a topic of intense research. Themotivation for this research is so far purely theoretical. The Standard ModelSU (3)×SU (2)×U (1) describes very accurately the physics at presently accessibleenergies. Present experimental results tell us that if supersymmetry is relevant inNature it has to be broken in the energy range explored so far. Nevertheless, thereare several theoretical reasons to expect that the Standard Model is not a completetheory. The main motivation for considering supersymmetry in the search for sucha theory is the improved convergence properties in the UV of supersymmetricfield theories. It is this feature which we eventually discuss in this chapter, afterintroducing the formalism of superfields.

UV divergences, which are softer than in non-supersymmetric theories, maybe crucial for solving the problem of the ‘naturalness’ of the Higgs sector,considered as one of the most serious defects of standard electroweak theory. Higgsparticles are scalar particles and their masses are subject to quadratic divergencesin perturbation theory. For example, one-loop radiative corrections to elementaryscalar masses are

δm2 = g2∫ � d4k

(2π)4

1

k2 − m= O

(α

π

)�2

Since the theory is renormalizable we can take the limit � → ∞ and as long aswe treat these masses as free parameters there is no reason to pay attention to themagnitude of δm2: bare parameters and corrections to them are not observablequantities. However, ultimately we want to understand the magnitude of thesemasses and then we must require

δm2 � O(m2)

509

510 15 Introduction to supersymmetry

Thus the quadratic divergence should be physically cut off at mass scale � ofthe order of the scalar particle mass and originating from new physics beyondthe Standard Model. The Standard Model requires the mass of the scalars to beO(100 GeV), since otherwise the couplings in the Higgs sector would be too largefor perturbation theory to be valid. So we need

� � 1 TeV

On the other hand if the Standard Model is a complete theory of everything butgravity the only cut-off which remains at our disposal is due to the fact that allparticles participate in the gravitational interactions. The energy at which gravityand quantum effects become of comparable strength can be estimated from the onlyexpression with the dimension of energy that can be formed from the fundamentalconstants c, h and GNewton. We have then

� ∼ MPlanck = c2(hc/GNewton)1/2 ≈ 1019 GeV

Two strategies have been proposed to resolve the above dilemma. One is topostulate that the scalar Higgs particle is composite with the compositeness scale� � 1 TeV which provides a natural cut-off in the quadratically divergent loops.This ‘technicolour’ scenario has been discussed in Chapter 11. Interesting as it isit cannot easily be reconciled with small fermion masses and with the magnitudeof the flavour-changing neutral interactions.

The other strategy is to cancel the loops noticing that the boson and fermionloops have opposite signs. This occurs in supersymmetric field theories and is partof the non-renormalization theorem to be discussed at the end of this chapter. Ofcourse, this cancellation cannot be exact since supersymmetry, if at all relevant,must be a broken symmetry. What we need, however, is

δm2 = +O(α/π)�2|bosons − O(α/π)�2|fermions � 1 TeV2

or effectively

|m2B − m2

F| � 1 TeV2

Thus, this strategy leads to the expectation that the supersymmetric partners ofknown particles have masses lighter than O(1 TeV).

The reader interested in studying supersymmetry at a level going beyond ourintroductory discussion is advised to consult, for example, Gates et al. (1983),Sohnius (1985), Wess & Bagger (1983), Nilles (1984) and references therein.

15.2 The supersymmetry algebra 511

15.2 The supersymmetry algebra

The continuous global symmetries of quantum field theory are generated byinfinitesimal transformations

δaϕ = i�[Qa, ϕ] (15.1)

where � is an infinitesimal parameter. The conserved charges Qa are expressed interms of fields in the way given by Noether’s procedure. The group of the Poincaretransformations with the Lorentz generators Mµv and the energy–momentumoperators Pµ satisfying the Poincare algebra (see Problem 7.1) and various groupsof internal symmetry transformations with generators transforming under theLorentz group as spin zero operators and forming a Lie algebra

[Qa, Qb] = icabc Qc (15.2)

are well known to be important symmetries of physical theories. Actually, Cole-man & Mandula (1967) have proved that all generators, other than the Poincaregroup generators, of symmetry transformations which form Lie groups with realparameters must have spin zero. Thus, any symmetry group of the S-matrixwhose generators obey well-defined commutation relations must commute withthe Poincare group. Since the Casimir operators of the Poincare group are themass square operator P2 = PµPµ and the generalized spin operator W 2 = WµWµ

where Wµ is the Pauli–Lubanski vector

Wµ = 12ε

µνρσ Pν Mρσ

one can conclude that all members of an irreducible multiplet of the internalsymmetry group must have the same mass and the same spin.

The Coleman–Mandula theorem, however, leaves open the possibility of includ-ing symmetry operations whose generators obey anticommutation relations. Sucha generalization of a Lie algebra is called a graded algebra or a superalgebra. Thegenerators of a graded Lie algebra consist of even elements Ai and odd elementsBα and have commutation rules of the form

[Ai , A j ] = ci jk Ak

[Ai , Bα] = giαβ Bβ

{Bα, Bβ} = hαβi Ai

(15.3)

We are interested in an extension of the Poincare algebra into a graded Liealgebra which contains the Poincare algebra as its subalgebra. This is easilyaccomplished if we introduce spinorial charge Q which is a Majorana spinor or


equivalently a left-handed Weyl spinor Qα (α = 1, 2), where

Q =(

Qα

Qα

), Q = (

Qα, Qα

)(15.4)

where Q = Q†γ 0 and Q†α = Qα (at this point we advise the reader to consult

Appendix A for the two-component notation). Thus we have

[Mµν, Qα] = 12(σµν)α

β Qβ

[Mµν, Qα] = − 12(σµν)

βα Qβ

}(15.5)

The anticommutator {Qα, Q†β} transforms under the Lorentz transformations as

( 12 ,

12) and, since Pµ is the only generator of the Poincare algebra in such a

representation, we must have

{Qα, Qβ} = 2(σµ)αβ Pµ (15.6)

The factor 2 is the normalization convention and the sign is determined by therequirement that the energy E = P0 should be a semi-positive definite operator

2∑α=1

{Qα, Q†α} = 2 Tr[σµPµ] = 4P0 (15.7)

The supersymmetry generators must commute with the momenta

[Qα, Pµ] = [Qα, Pµ] = 0 (15.8)

Indeed, the commutator of the Q with a Pµ could contain the Lorentz representa-tions (1, 1

2) and (0, 12). Since there are no (1, 1

2) generators present we get, as themost general possibility,

[Qα, Pµ] = C(σµ)αβ Qβ

and the adjoint equation

[Qβ , Pµ] = C∗(σµ)βα Qα

Therefore

[[Qα, Pµ], Pν] = CC∗(σµσν)αβ Qβ

and from the Jacobi identity and from [Pµ, Pν] = 0 we get C = 0 (a graded Liealgebra is defined by (15.3) and by the requirement that the graded Jacobi identitiesare fulfilled; the graded cyclic sum is defined just as the cyclic sum, except thatthere is an additional minus sign if two fermionic operators are interchanged).Finally, the anticommutator of Qα with Qβ , by Lorentz covariance, must be alinear combination of Poincare generators in the representations (0, 0) and (1, 0)

15.3 Consequences of the supersymmetry algebra 513

of the Lorentz group. In addition, due to (15.8) and the Jacobi identity they mustcommute with Pµ. Such operators do not exist and we are left with (15.5), (15.6),(15.8) and

{Qα, Qβ} = 0 (15.9)

which together with the Poincare algebra form the simplest supersymmetricalgebra which has the Poincare group as space-time symmetry. It is called N = 1supersymmetry, meaning that there is one two-spinor supercharge Qα. Haag,Łopuszanski & Sohnius (1975) have shown that only two-spinor supercharges areacceptable as generators of graded Lie algebras of symmetries of the S-matrixwhich are consistent with relativistic quantum field theory. Thus the only possibleextension of the N = 1 supersymmetry is to have N > 1 two-spinor superchargesQαi , i = 1, . . . , N , which carry some representation of the internal symmetry†

[Qαi , Tj ] = ci jk Qαk (15.10)

We shall limit our discussion to N = 1 supersymmetry which has been the mostextensively studied one. In this case the full symmetry group of physical theorycan be a direct product of N = 1 supersymmetry and of some internal symmetrygroup.

To conclude these considerations we note the four-component form of (15.6)(remember that in four-component notation Q is a Majorana spinor)

{Q, Q} = 2γ µPµ (15.11)

15.3 Simple consequences of the supersymmetry algebra

We notice that

[Qα, W 2] �= 0

[Qα, P2] = 0

}(15.12)

i.e. supersymmetry multiplets contain different spins but are always degeneratein mass. Supersymmetry transformations relate fermions to bosons as followsimmediately from the fact that the Qαs have spin one-half. Since in Nature wedo not observe mass degeneracy among particles of different spin, supersymmetrymust be broken either explicitly or spontaneously if it is relevant to physics.Spontaneous breaking is theoretically more appealing.

† In the N = 1 case the only non-trivially acting internal symmetry is a single U (1), known under the name ofR-symmetry: [Qα, R] = Qα, [Qα, R] = −Qα . Since under parity Qα ↔ Qα we must have R → −R, i.e.the U (1) symmetry group is chiral. In the four-spinor notation [Q, R] = iγ5 Q.


Fig. 15.1.

The vacuum structure of supersymmetric theories is of a special character. Itfollows immediately from (15.7) that if there exists a supersymmetrically invariantstate

Qα|0〉 = 0 and Qα|0〉 = 0 (15.13)

then obviously

Evac = 0 (15.14)

Since from (15.7) the spectrum of H is semi-positive definite, Evac = 0 impliesthat the supersymmetrically invariant state is always at the absolute minimum ofthe potential, i.e. it is the true vacuum state. Thus, if the supersymmetric stateexists, it is the ground state and supersymmetry is not spontaneously broken. Inthis respect supersymmetry is different from ordinary symmetries with which asymmetric state may exist without being the ground state. Only if a state invariantunder supersymmetry does not exist is supersymmetry spontaneously broken andfrom

Qα|0〉 �= 0 and/or Qα|0〉 �= 0 (15.15)

and from (15.7). It follows that

Evac > 0

Thus supersymmetry is unbroken if, and only if, Evac = 0. It is spontaneouslybroken if, and only if, the energy of the vacuum is greater than zero. Animportant fact is that a non-zero vacuum expectation value of a scalar field does notnecessarily mean that supersymmetry is spontaneously broken. This is illustratedin Fig. 15.1 which shows the case in which the expectation value of a scalarfield breaks an internal symmetry but does not break supersymmetry because thevacuum energy is zero.

15.4 Superspace and superfields for N = 1 supersymmetry 515

In the case of spontaneously broken supersymmetry one can prove an analogueof Goldstone’s theorem. Consider the state obtained by acting on the vacuum withQα (or Qα)

|�〉 = Qα|0〉 (or |�〉 = Qα|0〉) (15.16)

By the definition of spontaneously broken symmetry, |�〉 is not zero. It is a state ofodd fermion number and, because [Qα, H ] = 0, it is degenerate with the vacuum.A state with such a property is a state of a single massless fermion of momentum p,in the limit p → 0. So spontaneously broken supersymmetry requires the existenceof a massless fermion, the so-called Goldstone fermion or goldstino.

Finally let us construct a hermitean operator

Q = (1/√

2)(Qα + Q†α) (15.17)

for α = 1 or 2 and restrict our attention to the subspace of states with zero three-momentum p = 0.† In the space of states with p = 0 we have

Q2 = H (15.18)

and all states of non-zero energy are paired by the action of Q. Given any bosonstate |b〉 of non-zero energy E one gets a fermion state | f 〉 acting with Q on |b〉

Q|b〉 = E1/2| f 〉and vice versa

Q| f 〉 = E1/2|b〉However, the zero-energy states are annihilated by Q and therefore they are notpaired in this way. Any bosonic or fermionic state of zero energy (vacuum state)forms a one-dimensional supersymmetry multiplet.

15.4 Superspace and superfields for N = 1 supersymmetry

Superspace

To formulate a supersymmetric field theory we must work out representations ofsupersymmetry algebra on field operators. The supersymmetry transformationsmust act linearly on field multiplets and leave the action invariant. An elegantformalism in which N = 1 supersymmetry is manifest is the formalism ofsuperfields in superspace. To construct the superfield multiplets we first representthe supersymmetry algebra in terms of differential operators acting in superspace.

† For a detailed discussion of the supersymmetric spectrum, including massless states, see, for example, Sohnius(1985).


To do so we can imitate the construction of differential operators representing thePoincare generators in Minkowski space which can proceed as follows.

Take a quantum field �(x) which depends on the four coordinates xµ. We canconsider �(x) to have been translated from xµ = 0

�(x) = exp(ix · P)�(0) exp(−ix · P) (15.19)

or, differentially,

i[Pµ,�] = ∂µ� (15.20)

The important point is that this transformation is compatible with the multiplicationlaw

exp(ix · P) exp(iy · P) = exp[i(x + y) · P] (15.21)

which holds because the operators Pµ commute with each other.The differential version of a Lorentz transformation can be derived writing

�′(x) = exp(− 12 iω · M)�(x) exp( 1

2 iω · M)

= exp(ix ′ · P) exp(− 12 iω · M)�(0) exp( 1

2 iω · M) exp(−ix ′ · P) (15.22)

where ω · M = ωµν Mµν and x ′ is determined by the equation

exp(− 12 iω · M) exp(ix · P) = exp(ix ′ · P) exp(− 1

2 iω · M)

= exp[ix · (P − 12 iω · [M, P])] exp(− 1

2 iω · M)+ O(ω2) (15.23)

or explicitly, for infinitesimal ωµν and using (see Problem 7.1)

[Mµν, Pλ] = −i(gµλPν − gνλPµ)

we get

x ′λ = xλ + ωλνxν (15.24)

Finally, writing

exp(− 12 iω · M)�(0) exp( 1

2 iω · M) = exp(− 12 iω ·�)�(0) (15.25)

where the � is some matrix representation of the algebra of the Mµν which dependson the spin of �, and using (15.22) one has

�′(x) = exp(− 12 iω ·�)�(x ′) (15.26)

In the differential form we get

i[Mµν,�(x)] = (xµ∂ν − xν∂µ + i�µν)�(x) (15.27)

This result follows essentially from the multiplication law (15.23).


We can now repeat a similar construction for the case of the N = 1 supersym-metry algebra. The first observation is that the presence of the anticommutator inthe supersymmetry algebra calls for an extension of the Minkowski space into asuperspace including some Grassmann parameters, in which differential operatorswill act representing the supersymmetry algebra. This stems from the fact that thealgebra must be exponentiated into a group, i.e. the product of two group elementslike

exp(i�Qα) exp(iQα�α) (15.28)

where �α and �α are for the time being arbitrary parameters, must be a groupelement. This is indeed ensured if we take �α and �β to be anticommutingcomponents of two-spinors

{�α,�β} = {�α, �β} = {�α, �β} = 0 (15.29)

Since

[�α Qα, Qβ �β] = 2�α(σµ)αβ�

β Pµ

[�α Qα,�β Qβ] = [Qβ �

β , Qα�α] = 0

}(15.30)

we can write the product (15.28) as a group element using the Baker–Campbell–Hausdorf formula (see Problem 2.9) in which the multiple commutators vanish forgenerators of the supersymmetry algebra

exp(i�α Qα) exp(iQα�α) = exp[i(�α Qα + Qα�

α + i�ασµαβ�

β Pµ)] (15.31)

The commonly used choice for � and � is the ‘real’ superspace z = (xµ,�α, �α)

with �α = (�α)∗ which in the four-component notation corresponds to choosing�i , i = 1, . . . , 4, as a Majorana spinor.

As for the Poincare group we can discuss separately the Lorentz transformationsL which form a subgroup of the super-Poincare group and supertranslations whichform a coset manifold with respect to L . Any element of the super-Poincare groupcan be uniquely decomposed into a product

S(x,�, �) L(ω) (15.32)

where a finite supertranslation S(x,�, �) is parametrized as follows

S(x,�, �) = exp(ix · P + i�α Qα + iQα�α) (15.33)

The multiplication law for two successive supertranslations, analogous to (15.21),can be evaluated with the help of the Baker–Campbell–Hausdorf formulaand (15.30):

S(y, η, η) S(x,�, �) = S(x ′,�′, �′) (15.34)


wherex ′µ = xµ + yµ − i�ασµ

αβ ηβ + iηασµ

αβ�β

�′ = η +�

�′ = η + �

(15.35)

In complete analogy with (15.23), multiplication with the Lorentz group gives

L(ω) S(x,�, �) = S(x ′,�′, �′) L(ω) (15.36)

where, for infinitesimal transformations, using (15.5) we get

x ′λ = xλ + ωλνxν

�′α = �α − 14 iωµν(σµν)β

α�β

�′α = �α + 14 iωµν(σµν)

αβ�

β

(15.37)

A superfield �(x,�, �) is now defined as a function of the parameters � and� in addition to xµ such that (see (15.19))

�(x,�, �) = S(x,�, �)�(0, 0, 0)S−1(x,�, �) (15.38)

with the coordinate transformations given by (15.35), and

L(ω)�(x,�, �)L−1(ω) = exp(− 12 iω ·�)�(x ′,�′, �′) (15.39)

where for infinitesimal ω the parameters x ′,�′, �′ are given by (15.37). Expanding

�(x ′,�′, �′) = �(x,�, �)+ (y − i�ση + iησ�)(∂�/∂x)

+ η(∂�/∂�)+ η(∂�/∂�)+ · · ·= �(x,�, �)+ iy[P,�] + i[ηQ,�] + i[Qη, �] + · · · (15.40)

we get the differential form of the supersymmetry generators acting on a superfield�(x,�, �)

i[Pµ,�] = ∂µ�

i[ηα Qα,�] = ηα(∂/∂�α + iσµαβ�

β∂µ)�

i[Qα ηα, �] = ηα(∂/∂�α + i�βσµ

βα∂µ)�

(15.41)

Using (15.37) and (15.39) we also get

i[Mµν,�] = xµ∂ν− xν∂µ+ 12 i�α(σµν)α

β(∂/∂�β)− 12 i�β(σµν)

αβ (∂/∂�

α)+ i�µν

(15.42)If � does not have overall spinor indices we can factorize out η in the last twoequations of (15.41), for example,

i[Qα,�] = (∂/∂�α + iσµαβ�

β∂µ)� (15.43)


The important notion in supersymmetry is that of covariant derivatives. As usualone wants to generalize

∂

∂�α� → Dα�,

∂

∂�α� → Dα�

so that the quantities Dα�, and Dα� transform under supertranslations as the fieldsthemselves. (Note that this requirement is satisfied by ∂µ�.) Thus we require

{Dβ, Qα} = 0

{Dβ, Qα} = 0

}(15.44)

and similarly for Dβ . This gives

Dβ = ∂/∂�β − iσµββ�

β∂µ

Dβ = −∂/∂�β + i�βσµββ∂µ

}(15.45)

One can check that{Dα, Dβ} = {Dα, Dβ} = 0

{Dα, Dβ} = 2iσµαβ∂µ

}(15.46)

In the four-component notation we have

D =(

Dα

Dα

), � =

(�α

�α

), � = (�α, �α)

where

Dα = εαβ Dβ = ∂/∂�α − iσ µαβ�β∂µ (15.47)

(since ∂/∂�β = −εβα ∂/∂�α) and consequently

D = ∂/∂�− iγ µ�∂µ (15.48)

Superfields

We are now prepared to construct superfields as power series expansions in �α and�α. The most general superfield in the two-component notation reads (we denoteit by F(x,�, �) reserving the � for the chiral superfields)

F(x,�, �) = f (x)+�χ(x)+ �χ(x)+��m(x)+ ��n(x)+�σµ�vµ(x)

+��λ(x)+ ��λ(x)+��d(x) (15.49)

This is the most general superfield due to the vanishing of the square of eachcomponent of �α and �α. Remember that

�� = �α�α = εαβ�α�β, �� = −�α�α = −εαβ�

α�β


or in other words�α�β = − 1

2εαβ��

�α�β = 12ε

αβ��

}(15.50)

Assuming that the superfield has no overall Lorentz indices, this superfield containsfour complex scalar fields f (x), m(x), n(x) and d(x), two spinors χ(x) and λ(x)in the ( 1

2 , 0) representation and two unrelated spinors χ(x) and λ(x) in the (0, 12)

representation of the Lorentz group, and one complex vector field vµ(x). Thuswe have altogether 16 bosonic and 16 fermionic degrees of freedom (a degree offreedom is an unconstrained single real field).

The transformation laws under supersymmetric transformations for the compo-nents of F(x,�, �) can be obtained by comparing powers of � and � in theequation

δF = δηF + δηF = δ f (x)+�δχ(x)+ �δχ(x)+��δm(x)+ · · · (15.51)

where the l.h.s. of (15.51) is given by (15.41). Note that it follows from (15.41)that the variation of the highest component of the superfield must always be aspace-time derivative.

One can see that the general superfield yields the component multiplet( f, χ, χ,m, n, vµ, λ, λ, d) which is reducible under supertranslations. To getirreducible component multiplets one must impose some conditions on the su-perfields. These conditions must be covariant with respect to supersymmetrictransformations but otherwise there is no general rule concerning their choice.One widely used set of such conditions is that which defines a so-called chiral(left-handed) superfield �(x,�, �) with

Dα� = 0 (15.52)

and an antichiral (right-handed) superfield �(x,�, �) with

Dα� = 0 (15.53)

Using (15.45) and solving the first order partial differential equations we get

�(x,�, �) = exp(−i�βσµβα�

α∂µ)�(x,�) (15.54)

and

�(x,�, �) = exp(i�βσµβα�

α∂µ)ˆ�(x, �) (15.55)

The fields �(x,�) and ˆ�(x, �) have the expansions

�(x,�) = A(x)+ 2��(x)−��F(x) (15.56)

�(x, �) = A(x)+ 2�(x)�− ��F(x) (15.57)

15.5 Supersymmetric lagrangian; Wess–Zumino model 521

and each contains Weyl spinors of only one chirality. The chiral superfield�(x,�, �) is formally obtained from �(x,�) by the translation (15.54)

�(x,�, �) = �(xµ − i�βσµβα�

α, �) = A(x)− i�σµ�∂µ A(x)

− 14��∂2 A(x)+ 2��(x)+ i��∂µ�(x)σµ�−��F(x)

(15.58)

and the analogous expression also holds for �(x,�, �). In deriving (15.58) wehave used the relation

(�σµ�)(�σν�) = 12 gµν(��)(��) (15.59)

The transformation rules under supertranslation of the component fields A, � andF follow from the general formula (15.51) and read

δA = δη A + δη A = 2ηα�α

δ�α = −ηα F − i(σµη)α∂µ A

δF = −2i∂µ�σµη

(15.60)

We see explicitly that the component multiplet of the chiral superfield is irreducibleunder supertranslations. Let us also mention that the chiral superfield is notthe only one used in the construction of supersymmetric theories. In particular,supersymmetric gauge theories require vector superfields defined by the conditionV (x,�, �) = V †(x,�, �) that contain spin one bosons.

15.5 Supersymmetric lagrangian; Wess–Zumino model

Let us begin our considerations with the observation that the product of twosuperfields is again a superfield

F1(x,�, �)F2(x,�, �) = F3(x,�, �) (15.61)

This follows from the representation (15.41) of the superalgebra which is of firstorder in differential operators. From two superfields �1 and �2 of the samechirality one can construct another chiral superfield

�3 = �1�2 (15.62)

and from two superfields of different chirality – two vector superfields

VS = 12(�1�2 + �1�2)

VA = 12 i(�1�2 − �1�2)

}(15.63)


corresponding to the symmetrized and antisymmetrized product, respectively. The�3 in (15.62) is chiral due to the fact that D is a first-order differential operatorwhich obeys the Leibnitz rule

D(�1�2) = (D�1)�2 +�1 D�2 (15.64)

A very important chiral superfield is the one which, as we shall see, can beidentified with the kinetic part of the supersymmetric lagrangian. It reads

�K = 14�D2� (15.65)

where � ≡ �† and it is a left-handed chiral superfield since D2� is a left-handedchiral superfield (see (15.46))

D D2� = 0 (15.66)

and the product of two left-handed superfields is again a left-handed superfield.We will need the notion of integration over the Grassmann variables � and �.

This has been introduced in Section 2.5 and we just add the following definitions:∫d2� =

∫d�2 d�1,

∫d2� =

∫d�1 d�2 (15.67)

Therefore ∫d2�� =

∫d2�� = −2 (15.68)

We also note that the Dirac δ-functions are

δ(�α) = �α, δ(−�α) = −δ(�α)

δ(2)(�) = − 12��, δ(2)(�) = − 1

2��

}(15.69)

The crucial observation for constructing supersymmetric lagrangians is the factthat for the general scalar superfield (15.49) the integral

SD = 14

∫d4x d2� d2� F(x,�, �) (15.70)

and for the chiral superfield the integral

SF = 14

∫d4x d2��(x,�, �)+ h.c. (15.71)

are invariant under supertranslations. Indeed we see from (15.49) that∫d2� d2� F(x,�, �) = 4d(x) (15.72)

and we remember that the variation of d(x) under supertranslation is a four-derivative. Thus SD is an invariant. For chiral fields SD vanishes but the

∫d4x d2�

15.5 Supersymmetric lagrangian; Wess–Zumino model 523

alone gives an invariant result. Under∫

d4x we can use the expansion (15.56)since the transformation (15.54) in its infinitesimal form gives δ� which is afour-derivative. Thus

SF = 12

∫d4x F(x)+ h.c. (15.73)

where F(x) is the highest component of the chiral superfield, and SF is an invariantof supertranslations since δF(x) is also a four-derivative. (We always assumethat the fields fall off fast enough at infinity.) Of course the superfields in SD

and SF must in general arise as products of ‘single-particle’ superfields since thelagrangian has to contain terms at least bilinear in these elementary fields.

As the last step towards constructing the supersymmetric lagrangian let us dosome dimensional analysis. Clearly, from (15.7) and (15.28) each � and � carriesa mass dimension − 1

2 . Thus taking the canonical dimensions 1 and 32 for the

boson and fermion fields we conclude that the whole ‘single-particle’ superfield hasdimension 1. The derivatives Dα and Dα have dimension 1

2 each and from (15.68)the dimension of the measure d4x d2� d2� ≡ dz is −2 and the dimension of thechiral measure d4x d2� is −3. In order to have a renormalizable theory we haveto construct a dimensionless action built of single-particle superfields so that nonegative-dimension coupling constants are present.

We are now ready to write down the supersymmetric lagrangian for a chiralsuperfield � with complex field components A, �α and F describing four bosonicand four fermionic degrees of freedom. In four dimensions this is the smallestpossible number of degrees of freedom which can be contained in a superfieldsince any multiplet must contain a spinor which has at least two complex (Weyl) orfour real (Majorana) components. It is clear from our considerations that the mostgeneral action for � (the so-called Wess–Zumino action) with the superkinetic termquadratic in the field reads

SWZ = 14

∫d4x d2�

(18�D2�− 1

2 m�2 − 13 g�3

)+ h.c. (15.74)

(the term λ� can be eliminated by a redefinition of fields). The first term in (15.74)can also be written as d2� d2��.

In terms of the field components denoted as follows

Re A ≡ S, Im A ≡ P, Re F ≡ FS, Im F ≡ FP

and

� =(

�α

�α

)where � is the four-component Majorana spinor, one can derive from (15.74) the


following lagrangian density

L = 12

(∂µS∂µS + ∂µP∂µP + i�/∂� + F2

S + F2P

)− m(SFS + P FP + 12��)

− g[(S2 − P2)FS + 2S P FP + �(S − iγ5 P)�

]+ 4-div (15.75)

This is the lagrangian of the original Wess–Zumino supersymmetric model. Itgives the following equations of motion

∂2S = −m FS − 2g(SFS + P FP + 12��)

∂2 P = −m FP − 2g(SFp − P Fs − 12�iγ5�)

i/∂� = m� + 2g(S − iγ5 P)�

FS = mS + g(S2 − P2)

FP = m P + 2gS P

(15.76)

An important observation is that the equations for the fields FS and FP are purelyalgebraic and consequently the FS and FP can be eliminated from the lagrangianand from the equations of motion. We then get

L = 12(∂µS∂µS − m2S2)+ 1

2(∂µP∂µP − m2 P2)+ 12�(i/∂ − m)�

− mgS(S2 + P2)− g�(S − iγ5 P)� − 12 g2(S2 + P2)2 (15.77)

and correspondingly

(∂2 + m2)S = −mg(3S2 + P2)− 2g2S(S2 + P2)− g��

(∂2 + m2)P = −2mgS P − 2g2 P(S2 + P2)+ g�iγ5�

(i/∂ − m)� = 2g(S − iγ5 P)�

(15.78)

The elimination of the auxiliary fields FS and FP gives, however, a theory whichis supersymmetric only ‘on-shell’. The lagrangian (15.77) is invariant undertransformations obtained from the supersymmetry transformations by eliminationof the fields FS and FP , using their own equations of motion, and the on-shellalgebra closes only if the equations of motion of the dynamical fields hold. Thisis understandable by counting the off-shell and on-shell number of degrees offreedom for scalar and fermion fields.

15.6 Supersymmetry breaking

Supersymmetry has many appealing theoretical features, but cannot be an exactsymmetry of Nature: it predicts that fermions and scalars appear in supersymmetricmultiplets with all particles in the multiplet having the same mass, in obvious con-tradiction to experiment. The only possible explanation is that, if supersymmetric

15.6 Supersymmetry breaking 525

partners of known particles exist, they must be significantly heavier (they cannotbe weakly coupled and therefore escape detection because the structure of the cou-plings follows strictly from the constructions of the supersymmetric lagrangian).Therefore, one has to propose mechanisms of supersymmetry breaking, whichwould, however, preserve the main virtue of supersymmetry which is the absenceof quadratic divergences. The most natural and elegant choice is spontaneoussupersymmetry breaking, analogous to that described in Chapter 9: supersymmetryis broken by a non-symmetric ground state (the vacuum).

Before discussing specific examples, we consider the most general renormaliz-able lagrangian describing interaction of n chiral fields �i , i = 1, . . . , n:

L = 18�i D2�i − W (�i )+ h.c. (15.79)

where W (x1, . . . , xn) is an arbitrary polynomial of degree 3 of n variables

W (�1, . . . , �n) = λi�i + 12 mi j�i� j + 1

3 gi jk�i� j�k (15.80)

Auxiliary fields Fi may be eliminated by their (purely algebraic) equations ofmotion:

Fi = −∂W ∗

∂ A∗i

(15.81)

(in this and the following equations we introduce the function W =W (A1, . . . , An) which has the same dependence on the scalar fields as thesuperpotential does on the superfields).

In terms of the physical component fields Ai , ψi lagrangian (15.79) may be read(after dropping all total four-derivatives):

L = i∂µψi σµψi + ∂µ A∗

i ∂µ Ai

− 1

2

[∂2W

∂ Ai∂ A jψiψ j + h.c.

]− V (Ai , A∗

j ) (15.82)

where the scalar potential V is

V = F∗i Fi =

∣∣∣∣∂W

∂ Ai

∣∣∣∣2 (15.83)

In the simplest case, lagrangian (15.79) contains only one superfield � (comparewith (15.74)–(15.78) for the Wess–Zumino lagrangian). The condition V = 0 fora supersymmetric minimum of the potential has the form (we keep the linear termin the superpotential):

λ+ m A + g A2 = 0 (15.84)

Denoting

Re A ≡ S Im A ≡ P (15.85)


(15.84) gives (assuming that λ, m and g are real):

λ+ mS + g(S2 − P2) = 0

m P + 2gS P = 0

}(15.86)

Eqs. (15.86) can always be satisfied. The solution depends on the sign of theexpression m2 − 4gλ:

〈S〉 = − 1

2g[m ± (m2 − 4gλ)1/2]

〈P〉 = 0

for m2 − 4gλ ≥ 0 (15.87)

〈S〉 = − 1

2gm

〈P〉 = ± 1

2g(4gλ− m2)1/2

for m2 − 4gλ ≤ 0 (15.88)

For each of these solutions Vmin = 0, i.e. in the simplest version of the Wess–Zumino model supersymmetry remains unbroken. Moreover, shifting the fieldsS → S − 〈S〉, P → P − 〈P〉 one can eliminate the term λ� from the lagrangianand shift one of the minima to 〈S〉 = 〈P〉 = 0. It is worth noting that for λ, g �= 0the considered lagrangian does not possess any global symmetry, so such a shiftdoes not lead to any global symmetry breaking and no massless Goldstone bosonappears in the spectrum.

Spontaneous supersymmetry breaking is possible and, in some special cases,even unavoidable in models with more chiral superfields. It was shown(O’Raifeartaigh 1975) that at least three scalar superfields are necessary to breaksupersymmetry spontaneously in the absence of gauge fields. Consider a modelwith three chiral superfields with the following superpotential:

W = m�1�2 +�3(λ+ g�21) (15.89)

Therefore the scalar potential V :

V =3∑

i=1

F∗i Fi = |m A2 + 2g A1 A3|2 + m2|A1|2 + |λ+ g A2

1|2 (15.90)

The conditions for a supersymmetric minimum of the potential, Fi = 0, read

m A2 + 2g A1 A3 = 0 (15.91)

m A1 = 0 (15.92)

λ+ g A21 = 0 (15.93)

One sees that for λ �= 0 (15.91)–(15.93) cannot be simultaneously fulfilled.


The potential (15.90) is always positive and supersymmetry in this model isspontaneously broken.

It is very instructive to analyse the mass spectrum of the O’Raifeartaigh model.For simplicity we assume m2 > 2gλ, the opposite case may be easily analysed ina similar way. For m2 > 2gλ the minimum of the potential,

Vmin = λ2 (15.94)

is achieved for

〈A1〉 = 〈A2〉 = 0 (15.95)

and with arbitrary 〈A3〉. The potential possesses the so-called flat direction.Introduce a new parameter µ defined by

〈A3〉 = µ

2g(15.96)

(µ can always be chosen real by proper field redefinition). With this choice, theterm in the lagrangian

− 12 mi jψiψ j − gi jkψiψ j 〈Ak〉 + h.c. (15.97)

gives for the Weyl fermion fields the following mass matrix:

Mψ = µ m 0

m 0 00 0 0

(15.98)

with the mass eigenvalues

mψ1,2 = (m2 + µ2/4)1/2 ± µ/2

mψ3 = 0

}(15.99)

For scalar Si , Pi fields the potential (15.90) gives the following mass matrices:

M2S =

µ2 + m2 + 2gλ mµ 0mµ m2 00 0 0

M2

P = µ2 + m2 − 2gλ mµ 0

mµ m2 00 0 0

(15.100)


The physical masses are

m2S1,2

= µ2/2 + m2 + gλ± [(µ2/2 + gλ)2 + m2µ2]1/2

m2P1,2

= µ2/2 + m2 − gλ± [(µ2/2 − gλ)2 + m2µ2]1/2

m2S3= m2

P3= 0

(15.101)

As can be seen, mass matrices (15.98), (15.100) obtained in the presence ofsupersymmetry breaking fulfil the following condition:

M2S + M2

P = 2M2ψ (15.102)

instead of being simply equal M2S = M2

P = M2ψ , like in the unbroken phase.

Also, they do not commute and cannot be simultaneously diagonalized. Hence, themassive physical fermion and scalar fields are not degenerate and they are differentmixtures of the initial supermultiplets. Taking the trace of (15.102) one finds thevery important relation connecting the masses of the physical states:∑

fermion states

m2i =

∑boson states

m2i (15.103)

Since the number of physical fields with spin s is 2s + 1, we can rewrite (15.103)in the form:

S TrM2 ≡∑

s

(−1)s(2s + 1)m2s = 0 (15.104)

In the case of unbroken supersymmetry this condition is trivially fulfilled, as themasses of all particles inside each supersymmetric multiplet are equal and (15.104)reduces to ∑

(supermultiplets)

∑(states in supermultiplet)

(−1)s(2s + 1) = 0 (15.105)

which reflects the basic rule that the number of the fermionic and bosonic states inthe supermultiplets are the same.

We have derived (15.104) for the special case of the O’Raifeartaigh lagrangian,but similar equations hold generally for all models with spontaneously brokensupersymmetry, and also those including the gauge interactions. We give the proofhere only for a model defined by the lagrangian (15.79) generalized to the case ofcompletely arbitrary (not necessarily polynomial) superpotential W (�1, . . . , �n)

(Ferrara, Girardello & Palumbo 1979). The scalar potential given by (15.83)is semi-positive definite as a consequence of supersymmetry. As discussed inSection 15.3, supersymmetry is spontaneously broken if, and only if, Vmin > 0.


The minimum of the potential is given by the conditions

∂V

∂ Ai= ∂2W

∂ Ai∂ A j

∂W ∗

∂ A∗j

= 0

∂V

∂ A∗i

= ∂W

∂ A j

∂2W ∗

∂ A∗i ∂ A∗

j

= 0

(15.106)

The scalar mass matrix can be obtained by expanding the potential (15.83) aroundthe minimum:

V = Vmin + ∂2V

∂ Ai∂ A∗j

(Ai − 〈Ai 〉)(A∗j − 〈A j 〉∗)

+ 1

2

∂2V

∂ Ai∂ A j(Ai − 〈Ai 〉)(A j − 〈A j 〉)

+ 1

2

∂2V

∂ A∗i ∂ A∗

j

(A∗i − 〈Ai 〉∗)(A∗

j − 〈A j 〉∗) (15.107)

where 〈Ai 〉 are solutions to (15.106). Denoting the shifted scalar fields as

Ai − 〈Ai 〉 = Si + iPi (15.108)

one can rewrite (15.107) as

V = Vmin + ∂2V

∂ Ai∂ A∗j

[Si S j + Pi Pj − i(Si S j − Sj Pi )]

+ 1

2

∂2V

∂ Ai∂ A j[Si S j − Pi Pj + i(Si S j + Sj Pi )]

+ 1

2

∂2V

∂ A∗i ∂ A∗

j

[Si S j − Pi Pj − i(Si S j + Sj Pi )] (15.109)

Hence the diagonal mass matrix elements for scalars Si and pseudoscalars Pi are

(M2S)i i = ∂2V

∂ Ai∂ A∗i

+ 1

2

(∂2V

∂ Ai∂ A∗i

+ ∂2V

∂ A∗i ∂ A∗

i

)(M2

P)i i = ∂2V

∂ Ai∂ A∗i

− 1

2

(∂2V

∂ Ai∂ A∗i

+ ∂2V

∂ A∗i ∂ A∗

i

) (15.110)

Therefore, the trace over the scalar mass matrices is equal to

Tr(M2S +M2

P) = 2∂2V

∂ Ai∂ A∗i

= 2∂2W

∂ Ai∂ A j

∂2W ∗

∂ A∗i ∂ A∗

j

(15.111)

The trace (15.111) is identical, up to the overall factor 2, to the trace of the square


M†FMF of the fermion mass term which can be read off from (15.82)

(MF)i j = ∂2W

∂ Ai∂ A j(15.112)

Therefore, we have proven the general version of (15.102) and (15.104).It is worth noting that the existence of Fi �= 0 solutions to (15.106) implies (by

(15.112)) that also the equation

MFψ0 = 0 (15.113)

has a non-zero solution. Therefore, the existence of the SUSY breaking minimaof the scalar potential implies the existence of at least one massless fermion in thetheory, the so-called goldstino. In this way we also proved the supersymmetricgeneralization of the Goldstone theorem (Goldstone 1961).

Eq. (15.104) and its generalizations mean that models with spontaneous super-symmetry breaking, although very appealing theoretically (a limited number of freeparameters, a well-defined mass spectrum), are difficult to accept from the phe-nomenological point of view. Eq. (15.104) predicts that at least some of the scalarmasses in the theory should be smaller than the maximal fermion mass. Thereforea realistic model should be able to explain why we do not observe such light scalarparticles. An interesting hypothesis is that supersymmetry is broken spontaneouslyin a ‘hidden’ sector which communicates with the observable sector (describedby the supersymmetric extension of the Standard Model) only via gravitationalinteractions. Such models are invariant under local supersymmetry transformationsand also include gravitational interactions (the so-called supergravity models).†The only effect of spontaneous supersymmetry breaking in the hidden sector on thevisible sector is the presence of some additional, explicitly non-supersymmetric,terms in the matter lagrangian. The structure of such terms is very special: althoughthey explicitly break supersymmetry, they preserve its most important virtue: theabsence of the quadratic divergences in the loop diagrams. We call this a softexplicit breaking.

The dynamics of supergravity theories is very complicated and still remainsmostly unknown. A simplified approach is often employed: the supersymmetriclagrangian is directly extended by adding all possible terms which do not introducequadratic divergences to the theory (Girardello & Grisaru 1982). Any two- andtrilinear scalar couplings are allowed and, after including gauge interactions, alsomass terms for the gauginos (supersymmetric partners of the gauge bosons).Supergravity gives us the theoretical motivation for this procedure, even if at

† Operators of supersymmetry transformations anticommute with Lorentz transformations; local supersymme-try transformations anticommute with local Lorentz transformations, i.e. local Minkowski space transforma-tions. Hence it is natural that local supersymmetry must include (super)gravitational interactions.

15.7 Supergraphs and the non-renormalization theorem 531

present we are unable to calculate the magnitude of the additional terms and wemay only try to estimate (restrict) their size on the basis of the existing experimentaldata. Such effective low energy models with soft supersymmetry breaking appearto be fully acceptable phenomenologically. An important example is the so-calledMinimal Supersymmetric Standard Model, the minimal realistic supersymmetrictheory containing the Standard Model.

15.7 Supergraphs and the non-renormalization theorem

Perturbation theory in superspace may be developed as an extension of ordinaryperturbation theory. Our aim is to calculate superfield Green’s functions (z ≡(x,�, �))

〈0|T�(z1) . . . �(zk) . . . �(zk+1) . . . �(zn)|0〉 (15.114)

from which one can obtain the component-field Green’s functions by power seriesexpansion in �1, �1 . . . �n, �n . We use the generating functional technique withgenerating functionals defined in Section 2.6. For discussing the necessary counter-terms in particular, it is very convenient (see Section 4.2) to calculate the effectiveaction

�[� . . . � . . .] =∑

n

∫d4x1 . . . d4xn d2�1 . . . d2�n �(n)(z1, . . . , zn)�(z1) . . . �(zn)

=∑

n

∫ n∏i=1

[d4 pi

(2π)4

]d2�1 . . . d2�n(2π)4δ

( n∑i=1

pi

)× �(n)(p1, . . . , pn;�1, . . . , �n)�(p1,�1) . . . �(pn, �n)

(15.115)

where the �(n)s are the 1PI Green’s functions.Let us begin with several auxiliary remarks. The difference between Dα and

−∂/∂�α is a derivative term which does not contribute under the d4x integral.Thus, using the equivalence of integration over and differentiation with respect toGrassmann variables and our conventions (15.50) and (15.67) we can convert

D2 → 2d2� (15.116)

under the d4x integral. Similarly, we have

D2 → 2d2� (15.117)

For the operators D2 D2 and D2 D2 acting on chiral and antichiral fields, respec-tively, we can write

D2 D2� = [D2, D2]� = (−16∂2 − 8iDσµ∂µ D)� = −16∂2� (15.118)


and similarly

D2 D2� = −16∂2� (15.119)

Thus, for example∫d2�� =

∫d2�(−D2 D2/16∂2)� =

∫d2� d2� (−D2/8∂2)� (15.120)

and∫d2��2 =

∫d2��(−D2 D2/16∂2)� =

∫d2�(D2/2)[�(−D2/8∂2)�]

=∫

d2� d2��(−D2/8∂2)� (15.121)

where we have used the fact that D� = 0.The free-field part of the action (15.74) can now be written as follows

S0 = 14

∫d4x d2�( 1

8�D2�− 12 m�2)+ h.c.

=∫

d4x d2� d2� [ 18��− 1

8 m�(−D2/8∂2)�− 18 m�(−D2/8∂2)�]

=∫

d4x d2� d2� 116(�, �)A

(�

�

)(15.122)

where

A =(

m D2/4∂2 11 m D2/4∂2

)(15.123)

To derive the Feynman rules we shall use the generating functional technique.We define the functional derivative for the chiral fields by the equation

δ

δ�(x,�, �)

∫d4x ′ d2�′F[�(x ′,�′, �′)] = F ′[�(x,�, �)] (15.124)

This definition implies

δ�(x ′,�′, �′)δ�(x,�, �)

= 12 D2[δ(2)(�′ −�)δ(2)(�′ − �)δ(4)(x ′ − x)] (15.125)

(to get (15.124) integrate by parts using (15.125) and analogously for �.The generating functional of the theory is

W [J, J ] = exp(iSint[δ/iδ J, δ/iδ J ])W0[J, J ] (15.126)


where W0[J, J ] is the generating functional for the superfield Green’s functions:

W0[J, J ] =⟨0

∣∣∣∣T exp

[i∫

d4x d2� d2� 12(�, �)

(−(D2/4∂2)J

−(D2/4∂2) J

)]∣∣∣∣0⟩(15.126a)

and J and J are chiral sources. Thus

G(n),0�...�

(z1, . . . , zn) = (−i)n δ

δ J (z1)· · · δ

δ J (zn)W0[J, J ]|J= J=0 (15.127)

Using the path integral representation for the W0[J, J ]

W0[J, J ] =∫

D�D� exp

{i∫

dz

[116(�, �)A

(�

�

)+ 1

2(�, �)B

]}(15.128)

where dz = d4x d2� d2� and

B =(−(D2/4∂2)J

−(D2/4∂2) J

)

and performing the Gaussian integration over � and � one gets (see (2.81))

W0[J, J ] = exp

(−i∫

dz BT A−1 B

)(15.129)

where

A−1 =

− m D2

4(∂2 + m2)1 + m D2 D2

16∂2(∂2 + m2)

1 + m D2 D2

16∂2(∂2 + m2)− m D2

4(∂2 + m2)

(15.130)

Finally we have

W0[J, J ] = exp

[−2i

∫dz

(− J

1

∂2 + m2J + 1

2 J14 m D2

∂2(∂2 + m2)J

+ 12 J

14 m D2

∂2(∂2 + m2)J

)](15.131)


Using (15.125), (15.127), (15.131) and integrating by parts when necessary weobtain the free superpropagators. For instance

G(2),0��

(z1, z2) = −2i(∂/∂ J (z1))(∂/∂ J (z2))

×∫

dz3 dz4 J (z3)δ(4)(�34)G(x3 − x4)J (z4)

= −2i∫

dz3 dz4 [ 12 D2

1δ(z14)]δ(4)(�34)G(x3 − x4)[ 1

2 D22δ(z23)]

= −2i 12 D2

112 D2

2[δ(4)(�12)G(x1 − x2)] (15.132)

where

δ(zi j ) ≡ δ(4)(�i −� j )δ(4)(xi − x j ) ≡ δ(4)(�i j )δ

(4)(xi − x j )

and

(∂21 + m2)G(x1 − x2) = δ(4)(x1 − x2)

and

D21δ

(2)(�1 −�2) = −D21

12(�1 −�2)

2 = 2 exp[i(�1 −�2)σµ�1∂

(1)µ ]

D21δ

(2)(�1 − �2) = −D21

12(�1 − �2)

2 = 2 exp[−i�1σµ(�1 − �2)∂

(1)µ ]

}(15.133)

(remember that ∂/∂�α = −εαβ∂/∂�β). To obtain the component-field propaga-tors we can use (15.133), (15.54) and (15.55) to write

G(2),0��

(z1, z2) = 〈0|T �(x1,�1)ˆ�(x2, �2)|0〉 = −2i exp(i2�1σ

µ�2∂(1)µ )G(x1−x2)

(15.134)and then use (15.56) and (15.57) to expand in �1 and �2. Analogously, we get†

G(2),0�� (z1, z2) = 2im

D22 D2

2

16∂2D2

1[G(x1 − x2)δ(4)(�12)] (15.135)

= −2im D21[δ(4)(�12)G(x1 − x2)] (15.135a)

† D21[δ(4)(�12) f (x1 − x2)] = D2

2[δ(4)(�12) f (x1 − x2)].


In momentum space we thus get, for example,

G(x1 − x2) = − 1

(2π)4

∫d4 p

exp[−ip(x1 − x2)]

p2 − m2

G(2),0��

= 2i

p2 − m2exp[(2�1�2 −�1�1 +�2�2)

αβσµ

αβpµ] (15.136)

It is, however, more convenient to take as free superpropagators the following:†

G(2),0��

= 2i

p2 − m2δ(4)(�12)

G(2),0�� = 2i

p2(p2 − m2)14 m D2δ(4)(�12)

G(2),0��

= 2i

p2(p2 − m2)14 m D2δ(4)(�12)

(15.137)

where

D2 = D2(p,�1)

Dα(p,�) = ∂/∂�α − (σµ�)α pµ

Using this convention we associate the factors 12 D2

1 and 12 D2

2 present in (15.132)and 1

2 D22 and 1

2 D21 present in (15.135) with the chiral vertices rather than with the

propagators. Notice that these factors cancel the p−2 factors in the �� and ��

propagators (15.137)The vertices can be obtained from the general formula (15.126) where

Sint

[δ

iδ J,

δ

iδ J

]= − 1

12 g∫

d4x

[d2�

(1

iδ J

)3

+ d2�

(1

iδ J

)3](15.138)

Let us consider, for instance, the three-point Green’s function G(3)��(z1, z2, z3) in

the first order in g. The first term in (15.138) gives, up to the overall combinatorialfactor,

G(3)��(z1, z2, z3) = −Sint

[δ

iδ J

]δ

δ J (z1)

δ

δ J (z2)

δ

δ J (z3)W0[J, J ]

∼ −ig∫

d4x4 d2�4 G(2),0�� (z1, z4)G

(2),0�� (z2, z4)G

(2),0�� (z3, z4) (15.139)

where G(2),0�� are given by (15.135). It is now convenient to regroup the factors 1

2 D2

present in each G(2),0�� (due to the functional differentiation over δ J ) as follows:

use one 12 D2 to replace

∫d2�4 by d4�4, associate the other two 1

2 D2s with the

† Factors 2 in the propagators correspond to our normalization (15.74) appropriate for the real component fields,see (15.75).


Fig. 15.2.

Fig. 15.3.

�3 vertex and leave the remaining D2s for use in the adjacent vertices. Thus a �3

vertex has two factors 12 D2 acting on two of three propagators entering it. We then

get the following rules:

(i) free propagators are given by (15.137)

(ii) vertices are read from Sint with an extra 12 D2 (or 1

2 D2) for each chiral (or antichiral)superfield, but omitting one 1

2 D2 (or 12 D2) for converting d2� (or d2�) into d4�

(iii) for each vertex there is an integration over d4� and for each loop the usual integrationover d4k/(2π)4

(iv) to obtain the effective action we compute amputated 1PI diagrams i.e. we replace theexternal propagators by the appropriate superfield with a 1

2 D2 (or 12 D2) omitted at a

vertex for each external chiral (or antichiral) superfield

The counting of the 12 D2 factors becomes even more evident when we consider the

one-loop and two-loop diagrams in Fig. 15.2. In the first case after amputating theexternal lines we are left with two propagators, i.e. with four factors 1

2 D2 or 12 D2.

Two of them are used to convert two d2� integrals into d4� integrals and two areleft as vertex factors, in agreement with our general rules. In the second case theinternal vertices have two 1

2 D2 or 12 D2 factors each.

Let us use these rules to calculate the amplitudes corresponding to the 1PI


Fig. 15.4.

diagrams given in Fig. 15.3. Diagram (a), after integrating by parts, gives

�(�, �) ∼ −g2∫

d4 p

(2π)4

∫d4k

(2π)4

1

k2(p + k2)

∫d4�1 d4�2 �(−p,�1)

× �(p,�2)δ(4)(�12)D2

1 D22δ

(4)(�12)

∼ g2∫

d4 p

(2π)4

∫d4k

(2π)4

1

k2(p + k2)

∫d4��(−p,�)�(p,�)

(15.140)

We have used the fact that (see (15.133))

D21 D2

2δ(4)(�12)�1=�2 = 4

This diagram is logarithmically divergent and contributes to the wave-functionrenormalization constant for the superfield (

∫d4�� counterterm).

Writing down the amplitudes for Fig. 15.3(b) and (c) we immediately seethat they vanish. Indeed we end up with integrals

∫d4�� = 0 because the

integrand is chiral. Thus, we see that at the one-loop level there is no mass andcoupling-constant renormalization other than that induced by the wave-functionrenormalization, neither infinite nor finite. Expressed in terms of the componentfields this result means that the contributions to the mass renormalization constantof, for example, the diagrams in Fig. 15.4 generated by the lagrangian (15.77) (wenow use the complex field notation A ≡ S + iP) cancel each other. Actually thereis no infinite renormalization of the mass and the coupling-constant to any order ofperturbation theory. This is the non-renormalization theorem for chiral superfields.(In general, there is finite renormalization of these parameters in higher orders. Itvanishes when renormalizing at vanishing external momenta.)

The above-mentioned validity of the non-renormalization theorem to any orderof perturbation theory follows from the fact that, by integrating by parts the factors12 D2 or 1

2 D2 one-by-one, the contribution from any Feynman diagram to theeffective action can be written in the form∫ n∏

i=1

d4 pi d4� f (p1, . . . , pn)�(p1,�1) . . . �(pn,�n) (15.141)

i.e. it can be expressed as an integral over a single d4� and with the function


f (p1, . . . , pn) translationally invariant. Of course, our result (15.140) is the sim-plest example of the general rule (15.141). From (15.141) it follows in particularthat all vacuum-to-vacuum diagrams vanish because without any superfields theexpression is annihilated by the d4� integration. Thus, the normalization factorin (15.131) is one.

Appendix A

Spinors and their properties

Our conventions are:

h = c = 1, xµ = (t, x), pµ = (E, p), gµν = (1,−1,−1,−1),

x2 = gµνxµxν = t2 − x2, p = −i∂/∂x, pµ = i∂/∂xµ

A plane wave solution to the Schrodinger equation

i∂

∂t|p〉 = E |p〉 (A.1)

is 〈x|p, t〉 = exp(−ip · x) = �(t, x) = �(x). Space-time translation (change ofthe origin of the reference frame) x ′µ = xµ + εµ gives �′(x ′) = 〈x′|p〉 = �(x) =exp(−ip · x). Hence,

�′(x) = �(x − ε) = exp[−ip · (x − ε)] ≈ (1 + ipµεµ)�(x) (A.2)

Thus, the translation of the state |p〉 is described by the operator

|p′〉 = exp(iεµ pµ)|p〉 (A.3)

Indeed �′(x) = 〈x|p′〉 ≈ (1 + ipµεµ)�(x).

Lorentz transformations and two-dimensional representations of thegroup SL(2,C)

The Lorentz group is the group of transformations which leave invariant the squarex2 ≡ xµxµ ≡ gµνxµxν of the four-vector xµ. The infinitesimal form of thesetransformations reads

xµ′ ≈ xµ + ωµνxν (A.4)

539

540 Appendix A

where ωµν ≡ gµκωκν = −ωνµ. Identifying ωµν with the transformation

parameters, (A.4) can be rewritten in the form

xµ′ ≈ xµ − i

2ωκρ (Mκρ)

µνxν (A.5)

with the Lorentz group generators in the vector representation given by

(Mκρ)µν = i(gκµgρν − gρµgκ

ν) (A.6)

The generators Mµν satisfy the following commutation relations[Mκρ, Mµν

] = i(gκν Mρµ − gκµMρν − gρν Mκµ + gρµMκν) (A.7)

In this abstract (representation-independent) form the commutation relations (A.7)define the Lie algebra of the Lorentz group. In any representation, the finitetransformations U (ω) can be then written as

U (ω) = exp

(− i

2ωκρ Mκρ

)(A.8)

with Mκρ in the appropriate representation.The connection of the group SL(2,C) (the group of 2 × 2 complex matrices M

of determinant 1) to the Lorentz group is analogous to the connection of the groupSU (2) of the two-dimensional unitary unimodular matrices to the rotation groupin three dimensions (for example, Werle (1966)). It can be established through thePauli matrices

σ 1 =(

0 11 0

), σ 2 =

(0 −ii 0

), σ 3 =

(1 00 −1

)(A.9)

(note that σ iσ j = δi j + iεi jkσ k , σ 2σ iσ 2 = −σ i∗) in the following way. To everyspace-time point xµ we can assign a matrix

σ · x ≡3∑

µ=0

σµxµ =(

x0 + x3 x1 − ix2

x1 + ix2 x0 − x3

)(A.10)

where

σ 0 =(

1 00 1

)(A.11)

such that

det(σ · x) = x2 (A.12)

The transformation x → x ′ is defined by the equation

σ · x ′ = M(σ · x)M† det M = 1 (A.13)

Spinors and their properties 541

Since

det(σ · x ′) = x ′2 = det(σ · x) = x2 (A.14)

the transformation

x ′µ = �µν(M) xν (A.15)

is a Lorentz transformation. The group SL(2,C) is homomorphic to therestricted Lorentz group (i.e. its proper orthochronous subgroup) det� = +1,�0

0 ≥ 1: L↑+ = SL(2,C)/Z2 (because M and −M generate the same Lorentz

transformation).The group SL(2,C) has two inequivalent two-dimensional (fundamental) rep-

resentations

λ′α(x′) = Mα

β λβ(x) (A.16)

χ ′α(x ′) = (M†−1)α β χ β (x) (A.17)

where λ and χ are two-component complex vectors (dots are introduced to remindus that the indices α and α are the indices of the two different representationsand cannot be contracted) and x ′ is given by (A.15). Two other possible two-dimensional representations,

λ′α = (MT−1)αβ λβ ≡ λβ(M−1)βα (A.18)

and

χ ′α = (M∗)α β χβ ≡ χβ (M†)β α (A.19)

are, as can be easily checked, equivalent to the representations (A.16) and (A.17),respectively, via the unitary transformations

λα = εαβλβ χ α = χβεβα

λα = εαβλβ χα = χ βεβα

(A.20)

where epsilons are defined by εαβ = −εαβ = (iσ2)αβ , i.e.

ε12 = −ε21 = −ε12 = ε21 = 1

ε12 = −ε21 = −ε12 = ε21 = 1

}(A.21)

and satisfy

εαβεβγ = δ

γα

εαβεβγ = δαγ

(A.22)

542 Appendix A

Since matrices M are unimodular, the εs are SL(2,C) invariant tensors:

εαβ = Mαγ Mβ

δεγ δ εαβ = εγ δ Mγα Mδ

β

εαβ = εγ δ(M†−1)γ α(M†−1)δ β εαβ = (M†−1)α γ (M†−1)β δεγ δ

(A.23)

From (A.16), (A.18) and (A.17), (A.19), using anticommutativity of φ, λ, itfollows that the combinations

λ · φ ≡ λαφα = φαλα = −λαφα = φ · λ

η · χ ≡ ηβ χβ = χβ η

β = −ηβ χβ = χ · ψ

}(A.24)

are Lorentz-invariant.To make the connection between the SL(2,C) representations ((A.16), (A.17))

and the Lorentz group more explicit recall that the Lorentz transformations (A.8)can also be parametrized in terms of

N i± = 1

2(J i ± iK i ) (A.25)

where

J i = 12ε

i jk M jk, K i = M0i (A.26)

satisfy [J i , J j

] = iεi jk J k[K i , K j

] = −iεi jk J k[J i , K j

] = iεi jk K k

(A.27)

and [N i±, N j

±] = iεi jk N k

±[N i+, N j

−] = 0

}(A.28)

We see that the Lie algebra of the Lorentz group can locally be represented as adirect product of two (complexified) SU (2) algebras. However, the resulting twoSL(2,C) groups† are not independent. This follows from the fact that their groupparameters must be complex conjugate to each other:

exp

(−iω0i M0i − i

2ωi j Mi j

)≡

exp(−iξi K i − iηi J i

) = exp(−iζi N i

+ − iζ ∗i N i−)

(A.29)

with ζi ≡ ηi − iξi . The representations can be built by exploiting this decomposi-tion. Two obvious ways in which the commutation relations (A.27) can be satisfied

† A complexified SU (2) is just the SL(2,C) group.


are

r1(N i+) = 1

2σi , r1(N j

−) = 0 (A.30)

r2(N i+) = 0, r2(N j

−) = 12σ

j (A.31)

It is clear that the finite-dimensional (non-unitary) representations of the Lorentzgroup can be labelled by the pair (m, n), where m(m + 1) is the eigenvalueof the N i

+N i+ operator and n(n + 1) is the eigenvalue of the N i

−N i− operator.

Thus, the representations (A.30) and (A.31) correspond to the ( 12 , 0) and (0, 1

2)

representations. Since J i = N i++N i

−, we can identify the spin of the representationwith m + n. Using (A.25), (A.30) and (A.31) we have

r1(

exp(−iξi K i − iηi J i )) = exp

(− i

2σ i (ηi − iξi )

)(A.32)

r2(

exp(−iξi K i − iηi J i )) = exp

(− i

2σ i (ηi + iξi )

)(A.33)

Identifying (A.32) with Mαβ (acting on spinors λβ) we must identify (A.33) with

(M†−1)α β which acts on spinors χ β . Introducing two sets of matrices (σµ)αβ and(σ µ)αβ defined by

σµ = (I, σ i ), σ µ = (I,−σ i ) (A.34)

and satisfying

σµσ ν + σ νσ µ = σ µσ ν + σ νσµ = 2gµν (A.35)

the Lorentz group generators in representations ( 12 , 0) and (0, 1

2) can be writtencompactly as

r1(Mµν) = 12σ

µν ≡ i

4(σµσ ν − σ νσ µ)

r2(Mµν) = 12 σ

µν ≡ i

4(σ µσ ν − σ νσµ)

(A.36)

with the obvious assignment of spinor indices: (σµν)αβ and (σ µν)α β . Here σµν

and σ µν are two-dimensional matrices. We thus have

λ′α(x′) = {exp[−(i/4)ωµνσ

µν]}αβλβ(x) (A.37)

χ ′α(x ′) = {exp[−(i/4)ωµνσµν]}α β χ

β (x) (A.38)

As can be verified using (A.35), the matrices σµ and σ µ defined in (A.34) satisfythe following identities

12σ

κρσµ − 12σ

µσ κρ − i(σ κgρµ − σρgκµ) = 0

12 σ

κρσ µ − 12 σ

µσ κρ − i(σ κgµρ − σ ρgκµ) = 0

}(A.39)

544 Appendix A

Taking into account (A.16)–(A.19), (A.37), (A.38) and the explicit form of theLorentz group generators in the vector representation (A.6) the identities (A.39)turn out to be the infinitesimal forms of the relations

�νµMσµM† = σ ν

�νµM†−1σ µM−1 = σ ν

}(A.40)

Thus, σµ and σ µ are numerically invariant tensors of the Lorentz group providedthe index µ transforms according to the vector representation of O(1, 3). They aretherefore the Clebsch–Gordan coefficients which relate the representation ( 1

2 ,12) of

SL(2,C) to the vector representation of O(1, 3).From the completeness relation

(σµ)αβ(σµ)κρ = 2δραδ

κ

β(A.41)

one derives the Fierz transformation for anticommuting (Grassmann) Weylspinors†

(λψ)(χ η) = 12(λσ

µχ)(ψσµη) = − 12(λσ

µχ)(ησµψ) (A.42)

Other useful relations are

σµσ ν = gµν − iσµν

σ µσ ν = gµν − iσ µν

}(A.43)

and

εκα(σµ)αβεβδ = (σµ)δκ

εδβ(σµ)βαεακ = (σ µ)κδ

}(A.44)

from which (for anticommuting spinors λ, χ ) follows the identity

(χ σ µλ) = −(λσµχ) (A.45)

Finally, from the hermiticity of the σµs we have:

(σµ

αβ)∗ = σ

µ

βα ((σ µ)αβ)∗ = (σ µ)βα (A.46)

Four-component Dirac spinors are built from two Weyl spinors transforming as( 1

2 , 0) and (0, 12) representations of SL(2,C) (and carrying the same charges under

all other transformations):

� =(

λα

χ α

)� = (

χα, λα

) = �†γ 0 (A.47)

† For commuting (c-numbers) spinors one gets + sign on the r.h.s. of (A.42).


We have, for instance, λ ·χ+ λ · χ = ��, λ · χ−λ ·χ = �γ 5�, λσ µλ+χσµχ =�γ µ�. For χ ≡ λ one obtains in this way the Majorana spinors. Dirac matriceshave in this representation the following form:

γ µ =(

0 σµ

σ µ 0

)(A.48)

This is the so-called chiral (or Weyl) representation. The matrix γ 5, defined as

γ 5 ≡ iγ 0γ 1γ 2γ 3 (A.49)

has in this representation the form

γ 5 =(−I 0

0 I

)(A.50)

The standard Dirac representation

γ 0 =(

I 00 −I

), γ i =

(0 σ i

−σ i 0

), γ 5 =

(0 II 0

)(A.51)

is obtained with the help of the unitary transformation:

γµ

Dirac = U †γµ

WeylU, �Dirac = U †�Weyl, with U = 1√2

(I −II I

)(A.52)

It is also useful to introduce chiral Dirac spinors in the chiral representation

�L =(λα

0

), �R =

(0χ α

)(A.53)

which satisfy the relations

PL,R�L,R ≡ 1 ∓ γ 5

2�L,R = �L,R, PL,R�R,L = 0 (A.54)

These can be introduced with no restriction on the internal charges of λ and χ . Inaddition, every Dirac spinor can be decomposed as � = �L +�R.

Under the Lorentz transformation both, �Dirac and �L,R transform according to

� ′(�x) = exp

(− i

4ωρκσ

ρκ

)�(x) (A.55)

with

σρκ = i

2

[γ ρ, γ κ

](A.56)

546 Appendix A

which are four-dimensional matrices. However, the reducibility of the four-dimensional representation is manifest only in the Weyl form where

σρκ

4×4 =(σρκ 0

0 σ ρκ

)(A.57)

Note that we use the same symbol for four- and two-dimensional matrices σµν .

Solutions of the free Weyl and Dirac equations and their properties

We begin by recalling the properties of the free particle solutions of the masslessWeyl equation

i(σ µ)αβ∂µλβ(x) = 0 (A.58)

Writing the positive and negative frequency solutions as

λpos(x) = exp(−ik · x) a(k)

λneg(x) = exp(+ik · x) b(k)

}(A.59)

(lower case indices are understood) multiplying (A.58) by σ 0 from the left andrecalling the definitions (A.34) we get (remember E ≡ k0 = |k|)

σσσ · k|k|a(k) = −a(k) (A.60)

and the same equation for b(k). For k/|k| = (sin θ cosφ, sin θ sinφ, cos θ) thewell-known solutions to these equations read

a(k) = Na exp(iϕa)a−(k)

b(k) = Nb exp(iϕb)a−(k)

}(A.61)

where Na,b are normalization factors, and

a−(k) =(− sin(θ/2) exp(−iφ)

cos(θ/2)

)(A.62)

is normalized to unity.Thus, the solution λpos(x) describes a fermion state with the spin projection onto

the direction of its momentum (+k) equal to −1/2, i.e. with helicity −1/2. Thesolution λneg(x), being an eigenstate of the three-momentum operator p ≡ −i∂/∂xwith the eigenvalue −k, consequently describes a negative energy fermion withhelicity +1/2. In the Dirac sea interpretation it represents therefore the absence ofa negative energy particle with momentum −k and helicity +1/2, i.e. the presenceof a positive energy antiparticle with momentum +k and helicity again +1/2(because the absence of the spin antiparallel to −k is equivalent to the presence


of the spin antiparallel to k). This interpretation becomes more obvious in thesecond quantization language.

Similarly, for the solutions of the equation

i(σµ)αβ∂µχβ(x) = 0 (A.63)

we get

χpos(x) = exp(−ik · x)a′(k) = Na′ exp(−ik · x) exp(iϕ′a)a+(k)

χneg(x) = exp(+ik · x)b′(k) = Nb′ exp(+ik · x) exp(iϕ′b)a+(k)

}(A.64)

(upper case dotted indices are understood), where

a+(k) =(

cos(θ/2)sin(θ/2) exp(iφ)

)(A.65)

is normalized to unity and such that we have

σσσ · k|k|a

′(k) = a′(k) (A.66)

and the same relation for b′(k). Therefore, χpos(x) represents a +1/2 helicityparticle, whereas χneg(x) describes a −1/2 helicity antiparticle. Normalizationfactors N are determined by requiring that there are 2E particles per unit volume,i.e. that ∫

unit volumed3x λσ 0λ =

∫unit volume

d3x χσ 0χ = 2E (A.67)

leading to N = (2E)1/2 for all N . With this normalization we get

a(k)αa∗(k)β = b(k)αb∗(k)β = kµσµ

αβ

a′(k)αa′∗(k)β = b′(k)αb′∗(k)β = kµ(σµ)αβ

}(A.68)

which are the analogues of (A.80) for four-component spinors.In the four-component spinor language ((A.47)–(A.54)) (A.58) and (A.63) both

take the form

iγ µ∂µ�L,R = 0 (A.69)

supplemented with different conditions γ 5�L = −�L and γ 5�R = �R, respec-tively which in the Weyl representation are trivially satisfied with

�L =(λα

0

), �R =

(0χα

)(A.70)

Thus, for massless fermions chirality eigenstates are at the same time helicityeigenstates. Chiral spinors in the Dirac representation are obtained by (A.52).

548 Appendix A

The Dirac equation

(iγ µ∂µ − m)�(x) = 0 (A.71)

has four independent free-particle solutions (two of which correspond to positiveand two other to negative energies) given by

�(x)pos = exp(−ip · x) u(p, s), s = 1, 2

�(x)neg = exp(+ip · x) v(p, s), s = 1, 2

}(A.72)

where pµ = (E, p), E ≡ p0 = (p2 + m2)1/2 > 0 is the energy–momentum of theparticle and the s numbers its polarizations in a chosen basis (see below). From(A.71) we get

( /p − m)u(p, s) = 0, ( /p + m)v(p, s) = 0 (A.73)

The solutions of (A.73), valid in any representation of the Dirac matrices, read:

u(p, s) = /p + m

[2m(m + E)]1/2u(O, s)

v(p, s) = −/p + m

[2m(m + E)]1/2v(O, s)

(A.74)

In (A.74) u(O, s) and v(O, s) are four linearly independent solutions in the restframe of the particle in which pµ

rest = (m,O). Obviously, pµ = �µν pν

rest,with �µ

ν being the appropriate Lorentz boost. In the rest frame s = 1 (2)numbers the solutions with spin projection onto a chosen unit spin vector s =(sin θ cosφ, sin θ sinφ, cos θ) equal + 1

2 (− 12 ). In the Dirac representation (A.51)

of the γ µ matrices, the two linearly independent (and orthogonal) solutions withpositive energy read

u(O, 1) = N

a+(s)00

, u(O, 2) = N

a−(s)00

(A.75)

where N are normalization factors and two-component spinors a± are defined in(A.65) and (A.62). For negative energy solutions we have

v(O, 1) = N

00

−a−(s)

, v(O, 2) = N

00

a+(s)

(A.76)

(to match the Dirac sea interpretation, the solutions with + 12 and − 1

2 spinprojections are numbered in reversed order). Eqs. (A.74)–(A.76) are sufficientto explicitly construct four-component spinors for massive fermions in the Dirac


representation (see, for example, Haber (1993)). Using the transformation (A.52)one can also construct them in the chiral representation. The helicity eigenstates

�p

|p| u(p, h) = ±1

2u(p, h) (A.77)

can be constructed by specifying the directions of s and p to be parallel:σ pa+(s) = a+(s).

The spinors (A.74) are normalized as follows (u ≡ u†γ 0; cf. (A.47) and (A.48)):

u(p, s)u(p, s ′) = N 2δss′, u(p, s)v(p, s ′) = 0,

v(p, s)v(p, s ′) = −N 2δss′, v(p, s)u(p, s ′) = 0

}(A.78)

or, equivalently,

u†(p, s)u(p, s ′) = N 2 E

mδss′, v†(p, s)v(p, s ′) = N 2 E

mδss′ (A.79)

As for the massless chiral spinors we choose N = (2m)1/2 corresponding to 2Eparticles in a unit volume. By explicit calculation we then find∑

s=1,2

u(p, s)u(p, s) = /p + m∑s=1,2

v(p, s)v(p, s) = /p − m

(A.80)

One can also define projection operators onto the states with positive and negativeenergy

�±(p) ≡ ±/p + m

2m(A.81)

with properties �2±(p) = �±(p), �±(p)�∓(p) = 0, �+(p) + �−(p) = 1 and

satisfying

�+(p)�(x)pos = �(x)pos, �−(p)�(x)pos = 0,

�−(p)�(x)neg = �(x)neg, �+(p)�(x)neg = 0

}(A.82)

Projection operators on the state which in the rest frame has spin parallel (anti-parallel) to the direction s for positive (negative) energy solution reads

P(s) = 1 + γ 5/s

2(A.83)

where the four-vector sµ (s2 = −1, p · s = 0) is related to s by the appropriateLorentz boost: sµ = �µ

νsνrest, sν

rest = (0, s). Because of the reverse assignment(A.76) P(s) projects the positive energy solutions onto the states with spin in therest frame parallel to s and negative energy solutions onto the states with spin

550 Appendix A

antiparallel to s (which in the Dirac sea interpretation represent the antiparticlewith spin parallel to s).

A useful basis for the spin states is obtained by choosing the four-vector sµ inthe form:

sµ =( |p|

m,

E

m

p|p|)

(A.84)

which is obtained from the unit vector s = p/|p| by the boost relating the restframe of the particle to the frame where it has a momentum p. In this case P(s)projects the positive energy solutions onto the states with spin parallel to s, i.e. tothe momentum of the state and, for negative energy solutions, onto the states withspin antiparallel to s. Consequently, P(s) projects onto the positive helicity statesfor both signs of energy since for negative energy solutions the momentum of thestate (in the sense of the eigenvalue of the operator −i∂/∂x) is −p, i.e. parallel to−s. This is confirmed by noting that for sµ given by (A.84)

P(s)�±(p) = 1

2

(1 ± �� · p

|p|)�±(p) (A.85)

where

�� ≡(σσσ 00 σσσ

)(A.86)

is the spin operator. Thus, for the negative energy solutions the operator (A.85)projects onto the states which in the Dirac sea interpretation represent the states ofthe antiparticle with physical momentum +p and the spin parallel to p, i.e. ontothe positive helicity states of the antiparticle.

For the matrix elements of processes involving massive Majorana particlesexternal wave-functions are provided by the solutions of the free Dirac equation.†

Parity

Transformation of the space reflection is the change of the description of thephysical system from the one given in the right-handed coordinate frame to thedescription in the left-handed frame. Thus, xµ → xµ′ with x0′ = x0, x′ = −x.

A positive (negative) energy solution (A.59) of the Weyl equation (A.58) forleft-handed fields upon substituting x = −x′ becomes the positive (negative)energy solution of (A.63) (in the primed variables) for right-handed fields withk′ = −k. Similarly, the solutions of the equation for right-handed fields become

† Massless Majorana fermions are indistinguishable from massless Weyl fermions (there are just two statesconnected by C PT ).


in the reflected frame the solutions of the equation for the left-handed fields withreversed momentum. This follows from the the following relations:

(σ 0)αβa′(−k)β = i exp[i(ϕ′a − ϕa)]a(k)α

(σ 0)αβa(−k)β = i exp[i(ϕa − ϕ′a)]a′(k)α

}(A.87)

and similar relations for b′(−k) and b(−k).For the four-component Dirac spinors, the positive (negative) energy solutions

(A.72) of the Dirac equation (A.71) with momentum p and spin projection label supon substituting x = −x′ and multiplying by γ 0 become the positive (negative)energy solutions of the Dirac equation in the reflected frame with momentum −pand the same spin label s. This follows from the relations

u(−p, s) = γ 0u(p, s), v(−p, s) = −γ 0v(p, s) (A.88)

For helicity spinors, from (A.62) and (A.65), we get instead

u(−p,−h) = (−1)12−h exp(−2ihφ)γ 0u(p, h)

v(−p,−h) = (−1)12+h exp(2ihφ)γ 0v(p, h)

(A.89)

Time reversal

Transformation of the time reversal can be viewed as a change of the directionof the time axis of the reference frame. Thus, t ′ = −t , x′ = x. A state withmomentum k and helicity −1/2 (+1/2) described originally by a solution ((A.59),(A.64)) of the Weyl equation ((A.58), (A.63)) in the new frame should be describedby a solution (with the same sign of the energy) of the Weyl equation ((A.58),(A.63)) (in the primed variables) with momentum k′ = −k. The solutions in thenew frame λTα(x ′) and χ α

T (x′) are obtained from the original ones by the operations

λTα(x ′) ≡ exp(iγ )(iσ 1σ 3)αβ(λ�(x))β

χ αT (x

′) ≡ exp(iγ )(iσ 1σ 3)α β (χ�(x))β

}(A.90)

Substituting t = −t ′ and taking the complex conjugate brings back the exponentialfactors to the right form exp[−i(Et ′ − k′ · x′)] (exp[+i(Et ′ − k′ · x′)]) for positive(negative) energy solutions. The correct behaviour of the spinor factors followfrom the properties

(iσ 1σ 3)a(−k) = exp(2iϕa)a�(k), (iσ 1σ 3)a′(−k) = − exp(2iϕ′a)a′�(k) (A.91)

iσ 1σ 3 = iσ 1σ 3 =(

0 i−i 0

)(A.92)

and similar relation for b(k) and b′(k).

552 Appendix A

For the four-component Dirac spinors, the positive (negative) energy solutions(A.72) of the Dirac equation (A.71) with momentum p and spin projection label supon substituting t = −t ′, complex conjugating and multiplying by iγ 1γ 3 becomethe positive (negative) energy solutions of the Dirac equation in the reflected framewith momentum −p and opposite spin s ′ = −s. This follows from the relations

iγ 1γ 3u(−p, s) = i(−1)su�(p, 3 − s)

iγ 1γ 3v(−p, s) = −i(−1)sv�(p, 3 − s)

}(A.93)

Charge conjugation

Consider a charged field � transforming as � → exp(−iq�)� under theU (1) group of gauge transformations interacting with a classical electromagneticpotential Aµ(x). The positive energy solution† �pos(x) of the equation

[(∂ + iqeA)2 + m2]�pos(x) = 0 (A.94)

describes the behaviour of a particle carrying charge q in the vector potentialAµ(x). The negative energy solution �neg(x) cannot by itself have such aninterpretation. However, the function �c

pos(x) = exp(iγ )(�neg(x))� havingpositive energy satisfies the equation

[(∂ − iqeA)2 + m2]�cpos(x) = 0 (A.95)

and describes the behaviour of another particle (called the antiparticle) carryingcharge −q in the same potential. It transforms as �c

pos → exp(+iq�)�cpos under

the U (1) group. In the potential Acµ(x) ≡ −Aµ(x), the solution �c

pos(x) hasobviously (up to a constant phase factor) an identical space-time form as thesolution �pos(x) in the original potential Aµ(x). Thus, the simultaneous changeof a particle for its antiparticle and of the sign of the potential is a symmetry of thetheory because the resulting physical system behaves like the original one (theirwave-functions are identical). The same applies to a set of fields �i transformingas a representation R of a non-abelian gauge group G and interacting with thenon-abelian potential T a Aa

µ(x) (T a are the generators of the group) provided wesubstitute T a Aa

µ(x) → T a Acaµ (x) = −T a� Aa

µ(x). �cpos(x) transforms as R� under

the action of G.For left-handed Weyl fields λi , transforming as a representation R of the gauge

group, the positive energy solution of the equation

iσ · (∂ + ig AaT a) · λpos(x) = 0 (A.96)

describes a helicity h = −1/2 particle interacting with the external potential

† Strictly speaking, for energy of the solution to be well defined, the potential Aµ should not depend on time.


T a Aaµ(x). The negative energy solution of this equation, λi

neg(x), upon complexconjugation (and raising the spinor index) becomes the positive energy solution ofthe equation

iσ · (∂ − ig AaT a�) · λpos(x) = 0 (A.97)

and describes a helicity h = +1/2 antiparticle (λpos(x) transforms as R� under theaction of the gauge group). This confirms the interpretation given to the negativeenergy solutions of (A.58) and (A.63). Clearly, there is no choice of T a Aca

µ (x)in which this antiparticle would behave like the helicity h = −1/2 particle inthe original potential. The wave-function of a helicity h = −1/2 antiparticledenoted as λc transforming as R� under the gauge transformations is a positiveenergy solution of the equation

iσ · (∂ − ig AaT a�) · λcpos(x) = 0 (A.98)

Obviously, in the potential T a Acaµ (x) = −T a� Aa

µ(x) the wave-function of thehelicity h = −1/2 antiparticle is identical (up to a constant phase factor) to theone of the helicity h = −1/2 particle in the original potential T a Aa

µ(x). Thus, ifthe physical system consists of both sets of fields, λi and λc

i , the charge conjugationis its symmetry.

Positive energy solutions of the equation(i/∂ − g A/ aT a − m

)�pos(x) = 0 (A.99)

for the Dirac field describe two polarization states of a massive, spin 1/2 particleand transform as representation R under the action of the gauge group. The positiveenergy wave-function

�cpos(x) = exp(iγ )Cγ 0��

neg(x) = exp(iγ )C�Tneg(x) (A.100)

where �neg(x) is a negative energy solution of (A.99) and the matrix C is chosenso that

(Cγ 0)γ µ�(Cγ 0)−1 = −γ µ (A.101)

satisfies the equation† (i/∂ + g A/ aT a� − m

)�c

pos(x) = 0 (A.102)

and (having positive energy) can be interpreted as a wave-function of the corre-sponding antiparticle in the same potential AaT a . Obviously, �c

pos(x) transformsas R�. In the potential T a Aca

µ (x) = −T a� Aaµ(x) the solution �c

pos(x) has the samespace-time form as the solution �pos(x) in the original potential and, therefore,charge conjugation is a symmetry of the physical system.

† Of course �cneg(x) = exp(iγ )C�T

pos(x) satisfies (A.102) with negative energy.

554 Appendix A

The matrix C defined by (A.99) can be taken as

C = iγ 0γ 2 =(

0 −iσ 2

−iσ 2 0

)(A.103)

which, for example, in the Weyl representation is given by (cf. (A.20) and (A.47))

C =( {εαβ} 0

0 {εαβ})=

0 −1 0 01 0 0 00 0 0 10 0 −1 0

(A.104)

The matrix C satisfies the following relations:

C−1 = C† = CT = −C, C2 = −I,[C, γ 5

] = 0 (A.105)

Under the operation (A.100) a solution of (A.99) characterized by the momen-tum and spin four-vectors pµ and sµ, i.e. satisfying

�(x) =(ε /p + m

2m

)(1 + γ 5/s

2

)�(x) (A.106)

transforms into

�c(x) ≡ exp(iγ )C�T(x) = exp(iγ )(Cγ 0)

(ε /p + m

2m

)� (1 + γ 5/s

2

)�

��(x)

=(−ε /p + m

2m

)(1 + γ 5/s

2

)�c(x) (A.107)

where ε = ±1, i.e. the solution with negative energy, momentum −p and spinantiparallel to s is transformed into the solution with positive energy, momentum pand spin parallel to s etc. Therefore

CuT(p, s) = v(p, s)

C vT(p, s) = u(p, s)

}(A.108)

The same relations are valid for the helicity eigenstates. For a field with definitechirality

�(x) =(ε /p + m

2m

)(1 ± γ 5

2

)�(x) (A.109)

we get

�c(x) = exp(iγ )(Cγ 0)

(ε /p + m

2m

)� (1 ± γ 5

2

)�

��(x)

=(−ε /p + m

2m

)(1 ∓ γ 5

2

)�c(x) (A.110)

Appendix B

Feynman rules for QED and QCD and Feynmanintegrals

In this Appendix we collect Feynman rules used in Chapters 4, 5 and 8 for λφ4

theory, QED and QCD, respectively.

Feynman rules for the λ�4 theory:

propagatori

p2 − m2 + iε

loop integration∫

d4k

(2π)4

vertex −iλ, p + q + r + s = 0

symmetry factors S = 12

S = 16

S = 12

S = 14

vertex counterterm −iλ(Z1 − 1)(n)

555

556 Appendix B

mass counterterm −i(Z0 − 1)(n)m2

wave-function +i(Z3 − 1)(n) p2

renormalizationcounterterm

Feynman rules for QED

incoming electron with momentum k: u(k, s)

outgoing electron with momentum k: u(k, s)

outgoing positron with momentum k: v(k, s)

incoming positron with momentum k: v(k, s)

fermion propagatori

/p − m + iε

photon propagator−i

k2 + iε

[gµν − (1 − a)

kµkν

k2

]

fermion–fermion–photon vertex +ieγµ

fermionwave-functionrenormalizationcounterterm

i(Z2 − 1)(n) /p

fermion masscounterterm

−im(Z0 − 1)(n) ≡ iδm(n)

photonwave-functionrenormalizationcounterterm

−i(Z3 − 1)(n)

(k2gµν − kµkν)

vertex counterterm +ie(Z1 − 1)(n)γµ

Feynman rules and Feynman integrals 557

Fermion loop parametrization

= (−1)Tr

[i

/p − mγµ

i

/p − /k − mγν

](−e2)

Feynman rules for QCD

three-gluon coupling −gclmn[(r − p)µgλν

+(r − q)λgµν

+(q − p)νgλµ]

four-gluon coupling

(−ig2)[cabeccde(gµσ gνρ

− gµρgνσ )

+ cacecbde(gµνgρσ

− gµρgνσ )

+ cadeccbe(gµσ gνρ

− gµνgρσ )]

quark–quark–gluonvertex

−igγµ(T a)bc

Tr[T aT b] = 12δ

ab

ghost–ghost–gluonvertex

+gcabcrµ

gluon propagator covariant gauge:

− 1

2a(∂µ Aµ

a )2

−i

k2 + iεδab

[gµν − (1 − a)

kµkν

k2

]

558 Appendix B

Coulomb gauge: ∂µ Aµa − (nµ∂

µ)(nµ Aµa ) = 0, nµ = (1, 0, 0, 0)

−i

k2 + iεδab

[gµν − k · n(kµnν + kνnµ)− kµkν

(k · n)2 − k2

]axial gauge: nµ Aµ

a = 0, n2 = 0 or n2 < 0

−i

k2 + iεδab

[gµν − kµnν + kνnµ

k · n+ n2

(n · k)2kµkν

]

ghost propagator − i

k2 + iεδab

quark propagator δiki

/p − m + iε

symmetry factors S = 1

2!

S = 1

2!

S = 1

3!

For each fermion and ghost loop there is a minus sign. As for λ�4 and for QED,counterterms can be explicitly introduced into the Feynman rules. The completelist is given in Appendix C.

Dirac algebra in n dimensions

{γ µ, γ ν} = 2gµν (B.1)

gµµ = n (B.2)

γ µγµ = n (B.3)

γµ/aγ µ = (2 − n)/a (B.4)

γ µ/ab/γµ = 4a · b + (n − 4)/ab/ (B.5)

γ µ/ab/c/γµ = −2c/b//a − (n − 4)/ab/c/ (B.6)

Tr 1l = 4 Tr(odd γ ) = 0 Tr[γ µγ ν] = 4gµν (B.7)


In four dimensions (ε0123 = −ε0123 = 1)

γ 5 = γ5 = iγ 0γ 1γ 2γ 3 = iεαβγργ αγ βγ γ γ ρ/4! (B.8)

Tr[γ 5γ αγ βγ γ γ ρ] = +iεαβγρ Tr 1l (B.9)

γµγργν = (Sµρνσ + iεµρνσ γ5)γσ (B.10)

Sµρνσ = gµρgνσ + gρνgρσ − gµνgρσ (B.11)

The Dirac and chiral representations for γ s are given in Appendix A.

(γ 5)† = γ 5, (γ µ)† = γµ ={

γ µ µ = 0−γ µ µ ≥ 1

(B.12)

Feynman parameters

1

ab=∫ 1

0dx

1

[(1 − x)b + xa]2(B.13)

1

anb= n

∫ 1

0dx

xn−1

[(1 − x)b + xa]n+1(B.14)

1

abc= 2

∫ 1

0dx∫ 1

0dy

x

[axy + bx(1 − y)+ c(1 − x)]3

= 2∫ 1

0du∫ 1−u

0dw

1

[aw + b(1 − u − w)+ cu]3(B.15)

1

a1a2 . . . an= (n − 1)!

∫ 1

0dxn dxn−1 . . . dx2

× xn−2n xn−3

n−1 . . . x13 x0

2

[(1 − xn)an + xn[(1 − xn−1)an−1 + xn−1[· · · + x3[(1 − x2)a2 + x2a1]] . . .]n

(B.16)

Feynman integrals in n dimensions

I0 =∫

dn p

(2π)n

1

(p2 + 2k · p + M2 + iε)α

= i(−π)n/2

(2π)n

�(α − 12 n)

�(α)

1

(M2 − k2 + iε)α−n/2(B.17)

Iµ =∫

dn p

(2π)n

pµ

(p2 + 2k · p + M2 + iε)α= −kµ I0 (B.18)

560 Appendix B

Iµν =∫

dn p

(2π)n

pµ pν

(p2 + 2k · p + M2 + iε)α

= I0

[kµkν + 1

2 gµν(M2 − k2)1

α − 12 n − 1

](B.19)

Iµνρ =∫

dn p

(2π)n

pµ pν pρ

(p2 + 2k · p + M2 + iε)2

= −I0

[kµkνkρ + 1

2(gµνkρ + gµρkν + gνρkµ)(M2 − k2)1

α − 12 n − 1

](B.20)

Also, by convention ∫dn p (p2)−1 = 0

α-representation

1

p2 − m2 + iε= 1

i

∫ ∞

0dα exp[iα(p2 − m2 + iε)] (B.21)

Gaussian integrals∫d4k exp[i(ak2 + 2b · k)] = 1

i

(π

a

)2

exp

(−i

b2

a

)(B.22)

∫d4k kµ exp[i(ak2 + 2b · k)] = 1

i

(π

a

)2

exp

(−i

b2

a

)(−bµ

a

)(B.23)

∫d4k kµkν exp[i(ak2 + 2b · k)] = 1

i

(π

a

)2

exp

(−i

b2

a

)(iagµν + 2bµbν

2a2

)(B.24)

λ-parameter integrals∫ ∞

0

dλ

λexp[i(A + iε)λ] = − ln(A + iε)+∞ (B.25)∫ ∞

0

dλ

λ[exp(iAλ)− exp(iBλ)] exp(−ελ) = ln

B + iε

A + iε(B.26)∫ ∞

0

dλ

λ2exp(−ελ) f (λ) =

ε→0

∫ ∞

0

dλ

λexp(−ελ)

d f (λ)

dλ(B.27)


Feynman integrals in light-like gauge n · A = 0, n2 = 0

Using

1

aαbβ= �(α + β)

�(α)�(β)

∫ 1

0dx

xα−1(1 − x)β−1

[ax + b(1 − x)]α+β(B.28)

one can show that

I (α) =∫

dnk

(2π)n

1

(k2 − 2p · k + M2)α(k · n)β

= 1

(p · n)β

∫dnk

(2π)n

1

(k2 − 2p · k + M2)α(B.29)

Iµ(α) =∫

dnk

(2π)n

kµ

(k2 − 2p · k + M2)α(k · n)β

= I (α)

[pµ − β

2(α − 12 n − 1)

(M2 − p2)nµ

p · n

](B.30)

Iµν(α) =∫

dnk

(2π)n

kµkν

(k2 − 2p · k + M2)α(k · n)β

= I (α)

[12(M2 − p2)

gµν

α − 1 − 12 n

+ pµ pν − pµnν + pνnµ

p · n(M2 − p2)

β

2(α − 1 − 12 n)

+ nµnν

(p · n)2(M2 − p2)2 β(β + 1)

4(α − 1 − 12 n)(α − 2 − 1

2 n)

](B.31)

Convention for the logarithm

The logarithm has a cut along the negative real axis (ln z = ln |z| + i arg z, −π <

arg z < π ). With this convention the rule for the logarithm of a product is

ln(ab) = ln a + ln b + η(a, b) (B.32)

where

η(a, b) = 2π i[�(− Im a)�(− Im b)�(+ Im ab)

−�(Im a)�(Im b)�(− Im ab)] (B.33)

Important consequences are:

(i) ln(ab) = ln a + ln b if Im a and Im b have different signs;(ii) ln(a/b) = ln a − ln b if Im a and Im b have the same sign;

(iii) if a and b are real then

ln(ab + iε) = ln(a + i(ε/b))+ ln(b + i(ε/a))

562 Appendix B

The integration∫ 1

0 [dx/(ax + b)] for arbitrary complex a and b requires some care.Naively we have a−1 ln(ax + b) but if for 0 < x < 1 the argument can be real andnegative, one must actually split the integral into two intervals. It is easier to firstdivide out a:

1

a

∫ 1

0

dx

x + b/a= 1

aln

(x + b

a

)∣∣∣∣10

(B.34)

No matter what the sign is of Im b/a the argument of ln never crosses the cut forreal x . Thus the answer is (1/a) ln[(a+b)/b] since the imaginary parts of (1+b/a)and b/a have the same sign.

Spence functions:

F(ξ) =∫ ξ

0

ln(1 + x)

xdx (B.35)

F(ξ)+ F(1/ξ) = 16π

2 + 12 ln2 ξ (B.36)

F(−ξ)+ F(ξ − 1) = − 16π

2 + ln ξ ln(1 − ξ) (B.37)

F(1) = 112π

2, F(−1) = − 16π

2 (B.38)

F(ξ) = ξ − 14ξ

2 + 19ξ

3 + · · · (B.39)

L(x) =∫ x

0

ln(1 − u)

udu = F(−x) (B.40)

L(x) =∫ x

0

ln |1 − u|u

du ± iπ ln x (B.41)

Indefinite integrals∫(dz/z) ln(z − z0) = 1

2 ln2 z − L(z0/z) (B.42)∫(dz/z) ln(z + z0) = ln z ln z0 + F(z/z0) (B.43)

Appendix C

Feynman rules for the Standard Model

We use the notation introduced in Chapter 12. Moreover, y Af ( f = l, u, d) stand

for eigenvalues of the corresponding renormalized Yukawa coupling matrices Y ACf .

Obviously, we have y Af ≡ (e/

√2sc)(m f A/MZ ) where quantities on the r.h.s. are

the renormalized lagrangian parameters. Since there is no flavour mixing in thelepton sector we can also write Y AC

l = Y Al δAC = y A

l δAC , δY ACl = δY A

l δAC =δy A

l δAC .The QCD renormalization constants ZS , ZG , ZηG are, respectively, Z1YM, Z3, Z2

of Chapter 8 (we use the scheme in which g = g = ˜g; see the comment following(8.3)). The ZηL and ZηY denote renormalization of ghost fields associated with theSUL(2) and UY(1) gauge fields, respectively. The gauge fixing parameter in QCDis a and in the electroweak theory it is denoted by ξ . Wherever more convenient weuse the explicit splitting of the renormalization constants Zi into Z = 1+ δZi . Forone-loop calculations all counterterms can be approximated by expressions linearin δZi s and δYi s.

Propagators of fermions

Leptons

i

/pPL δAB

iδZ Al /pPLδ

AB

i

/p − meA

δAB

i(δZ A

l /pPL + δZ Ae /pPR

)− i√

2

[(v − δv)(y A

l + δy Al )− y A

l v]δAB

563

564 Appendix C

Quarks

i

/p − mu A

δABδi j

i[δZ AB

q /pPL + δZ ABu /pPR

]δi j

− i√2

[(v − δv)(y A

u δAB + δY ABu )PL

+ (v − δv)(y Au δAB + (δY †

u )AB)PR

− vy Au δAB

]δi j

i

/p − mdA

δABδi j

i[(V †δZq V )AB /pPL + iδZ AB

d /pPR]δi j

− i√2

[(v − δv)(y A

d δAB + δY ABd )PL

+ (v − δv)(y Ad δAB + (δY †

d )AB)PR

− vy Ad δAB

]δi j

Propagators of the gauge bosons

−i

k2 − M2W

[gµν − (1 − ξ)

kµkν

k2 − ξ M2W

]− iδZW

(gµνk2 − kµkν

)+ iM2

W

[ZH

(1 − δv

v

)2

Z2g Z−2

W − 1

]gµν

−i

k2 − M2Z

[gµν − (1 − ξ)

kµkν

k2 − ξ M2Z

]−i(c2δZW + s2δZ B)

(gµνk2 − kµkν

)+iM2

Z

[ZH

(1 − δv

v

)2

×(c4 Z2g Z−2

W + s4 + 2c2s2 Zg Z−1W

)− 1

]gµν

Feynman rules for the Standard Model 565

−i

k2

[gµν − (1 − ξ)

kµkν

k2

]−i(c2δZ B + s2δZW )

(gµνk2 − kµkν

)+is2c2 M2

Z ZH

(1 − δv

v

)2 (1 − Zg Z−1

W

)2gµν

−isc(δZW − δZ B)(gµνk2 − kµkν

)+iscM2

Z ZH

(1 − δv

v

)2 (s2 + c2 Zg Z−1

W

)×(Zg Z−1

W − 1)gµν

−i

k2

[gµν − (1 − a)

kµkν

k2

]δab

−iδZG(gµνk2 − kµkν

)δab

Propagators of the Higgs and Goldstone bosons

i

p2 − M2h

iδZH p2 − i

{δm2

+ 3

2

[(λ+ δλ)

(1 − δv

v

)2

− λ

]v2

}

i

p2 − ξ M2Z

iδZH p2 − i

{δm2

+ 1

2

[(λ+ δλ)

(1 − δv

v

)2

− λ

]v2

}

566 Appendix C

i

p2 − ξ M2W

iδZH p2 − i

{δm2

+ 1

2

[(λ+ δλ)

(1 − δv

v

)2

− λ

]v2

}

−i

{vδm2 + 1

2

[(λ+ δλ)

(1 − δv

v

)3

− λ

]v3

}

Propagators of the ghost fields

−iδab

p2

−iδZηGδab p2

−i

p2 − ξ M2Z

−i(c2δZηL + s2δZηY )p2

+iξ M2Z

{ZH

(1 − δv

v

)2[s4 ZηY + c4 ZηL Z2

g Z−2W

+ s2c2 Z1/2ηY

Z1/2ηL

Zg(Z−3/2W Z1/2

B + Z−1/2W Z−1/2

B )]−1

}−i

p2

−i(s2δZηL + c2δZηY )p2

+is2c2 M2Z ZH

(1 − δv

v

)2 [ZηY + ZηL Z2

g Z−2W

− Z 1/2ηY

Z1/2ηL

Zg(Z−3/2W Z1/2

B + Z−1/2W Z−1/2

B )]


−ics(δZηL − δZηY )p2

+icsξ M2Z ZH

(1 − δv

v

)2

×[s2(

Z1/2ηY

Z1/2ηL

Zg Z−1/2W Z−1/2

B − ZηY

)+ c2 Zg Z−1

W

(ZηL Zg Z−1

W − Z 1/2ηY

Z1/2ηL

Z−1/2W Z1/2

B

)]

−i

p2 − ξ M2W

−iδZηL p2 + iξ M2W

[ZH ZηL Z2

g Z−2W

(1− δv

v

)2

− 1

]

Mixed propagators (only counterterms exist)

−scMZ ZH

(1 − δv

v

) (Zg Z−1

W − 1)

pµ

−MZ

[ZH

(1 − δv

v

) (s2 + c2 Zg Z−1

W

)− 1

]pµ

∓iMW

[ZH

(1 − δv

v

)Zg Z−1

W − 1

]pµ

Gauge interactions of fermions

Leptons

−ie√2s

γµδAB PL

−ie√2s

γµ(Z Al Zg Z−1

W − 1)δAB PL

568 Appendix C

−ie

2scγµδ

AB PL

−ie

2scγµ

[Z A

l (c2 Zg Z−1W + s2)− 1

]δAB PL

−ie

2γµZ A

l (Zg Z−1W − 1)δAB PL

+ie

2scγµδ

AB[(1 − 2s2)PL − 2s2 PR

]+i

e

2scγµδ

AB{[

Z Al (c2 Zg Z−1

W − s2)

− (1 − 2s2)]PL − 2s2δZ A

e PR}

+ieγµδAB

+ieγµδAB{[

12 Z A

l (1 + Zg Z−1W )− 1

]PL + δZ A

e PR}

Quarks

−igsγµT ai jδ

AB

−igsγµT ai j

[(Z AB

q ZS Z−1G − δAB

)PL

+ (Z AB

u ZS Z−1G − δAB

)PR]

−igsγµT ai jδ

AB

−igsγµT ai j

{[(V † Zq V )AB ZS Z−1

G − δAB]PL

+ (Z AB

d ZS Z−1G − δAB

)PR}


−ie√2s

γµV ABδi j PL

−ie√2s

γµ

[(Zq V )AB ZS Z−1

G − V AB]δi j PL

−ie√2s

γµ(V†)ABδi j PL

−ie√2s

γµ

[(V † Zq)

AB ZS Z−1G − (V †)AB

]δi j PL

−i 23 eγµδ

ABδi j

−i 23 eγµ

{[14 Z AB

q (3Zg Z−1W + 1)− δAB

]PL

+ δZ ABu PR

}δi j

+i 13 eγµδ

ABδi j

+i 13 eγµ

{[12(V

† Zq V )AB(3Zg Z−1W − 1)− δAB

]PL

+ δZ ABd PR

}δi j

−ie

2scγµ

[(1 − 4

3 s2)PL − 43 s2 PR

]δABδi j

−ie

2scγµ

{[Z AB

q (c2 Zg Z−1W − 1

3 s2)

− (1 − 43 s2)δAB

]PL − 4

3 s2δZ ABu PR

}δi j

+ie

2scγµ

[(1 − 2

3 s2)PL − 23 s2

W PR]δABδi j

+ie

2scγµ

{[(V † Zq V )AB(c2 Zg Z−1

W + 13)

− (1 − 23 s2)δAB

]PL − 2

3 s2δZ ABd PR

}δi j

570 Appendix C

Yukawa interactions of fermions

Leptons

−iy Al δAB PL(R)

−iδy Al δAB PL(R)

− i√2

y Al δAB

− i√2δy A

l δAB

+ 1√2

y Al γ 5δAB

+ 1√2δy A

l γ 5δAB

Quarks

− i√2

y Au δABδi j

− i√2

(δY AB

u PL + (δY †u )

AB PR)δi j

− i√2

y Ad δABδi j

− i√2

(δY AB

d PL + (δY †d )

AB PR

)δi j

− 1√2

y Au γ 5δABδi j

+ 1√2

(δY AB

u PL − (δY †u )

AB PR)δi j


+ 1√2

y Ad γ 5δABδi j

− 1√2

(δY AB

d PL − (δY †d )

AB PR

)δi j

−i(y A

d PL − yBu PR

)(V †)ABδi j

−i[(δYd V †)AB PL − (V †δY †

u )AB PR

]δi j

+i(y A

u PL − yBd PR

)V ABδi j

+i[(δYu V )AB PL − (V δY †

d )AB PR

]δi j

Gauge interactions of the gauge bosons

All momenta are outgoing.

+gscabc[gσµ(p1 − p2)

ν + gµν(p2 − p3)σ

+ gνσ (p3 − p1)µ]

+gsδZScabc[gσµ(p1 − p2)

ν

+ gµν(p2 − p3)σ + gνσ (p3 − p1)

µ]

−ie[gσµ(p1 − p2)


+ gνσ (p3 − p1)µ]

−ieδZg[gσµ(p1 − p2)

ν

+ gµν(p2 − p3)σ + gνσ (p3 − p1)

µ]

572 Appendix C

−iec

s

[gσµ(p1 − p2)


+ gνσ (p3 − p1)µ]

−ieδZgc

s

[gσµ(p1 − p2)

ν

+ gµν(p2 − p3)σ + gνσ (p3 − p1)

µ]

−ig2s [cabececd(gµσ gνλ − gµλgνσ )

+ cacecedb(gµλgνσ − gµνgλσ )

+ cadecebc(gµνgλσ − gµσ gνλ)]

−ig2s (Z2

S Z−1G − 1)

[cabececd(gµσ gνλ − gµλgνσ )

+ cacecedb(gµλgνσ − gµνgλσ )

+ cadecebc(gµνgλσ − gµσ gνλ)]

−ie2[2gµνgσλ − gµλgνσ − gµσ gνλ

]−ie2

(Z2

g Z−1W − 1

) [2gµνgσλ − gµλgνσ − gµσ gνλ

]

−ie2 c2

s2

[2gµνgσλ − gµλgνσ − gµσ gνλ

]−ie2 c2

s2

(Z2

g Z−1W − 1


]

−ie2 c

s

[2gµνgσλ − gµλgνσ − gµσ gνλ

]−ie2 c

s

(Z2

g Z−1W − 1


]


+ie2

s2

[2gµλgσν − gµνgλσ − gµσ gνλ

]+i

e2

s2

(Z2

g Z−1W − 1

) [2gµλgσν − gµνgλσ − gµσ gνλ

]

Self-interactions of the Higgs and Goldstone bosons

−i3vλ

−i3 [(λ+ δλ)(v − δv)− λv]

−ivλ

−i [(λ+ δλ)(v − δv)− λv]

−i3λ

−i3δλ

−iλ

−iδλ

−iλ

−iδλ

574 Appendix C

−i2λ

−i2δλ

Gauge interactions of the Higgs and Goldstone bosons

+ie

scMZ gµν

+ie

scMZ

[ZH

(1 − δv

v

)(c4 Z2

g Z−2W

+ 2s2c2 Zg Z−1W + s4

)− 1

]gµν

+iescMZ ZH

(1 − δv

v

) (Zg Z−1

W − 1)2

gµν

+ieMZ ZH

(1 − δv

v

) (Zg Z−1

W − 1)

× (s2 + c2 Zg Z−1

W

)gµν

+ie

sMW gµν

+ie

sMW

[ZH Z2

g Z−2W

(1 − δv

v

)− 1

]gµν


−ies MZ gµν

−ies MZ

[ZH Zg Z−1

W

(1 − δv

v

)− 1

]gµν

+iecMZ gµν

+iecMZ

[ZH Zg Z−1

W

(1 − δv

v

)− 1

]gµν

∓ie

2sc(1 − 2s2)(p1 + p2)µ

∓ie

2sc

[ZH(c

2 Zg Z−1W − s2)− (1 − 2s2)

]×(p1 + p2)µ

∓ie(p1 + p2)µ

∓ie

2

[ZH(1 + Zg Z−1

W )− 2](p1 + p2)µ

− e

2sc(p1 + p2)µ

− e

2sc

[ZH(c

2 Zg Z−1W + s2)− 1

](p1 + p2)µ

−e

2ZH(Zg Z−1

W − 1)(p1 + p2)µ

576 Appendix C

±ie

2s(p1 + p2)µ

±ie

2s

(ZH Zg Z−1

W − 1)(p1 + p2)µ

− e

2s(p1 + p2)µ

− e

2s

(ZH Zg Z−1

W − 1)(p1 + p2)µ

+ie2

2s2c2gµν

+ie2

2s2c2

[ZH(c

4 Z2g Z−2

W

+ 2s2c2 Zg Z−1W + s4)− 1

]gµν

+ie2

2ZH

(Zg Z−1

W − 1)2

gµν

+ie2

2scZH

(Zg Z−1

W − 1) (

c2 Zg Z−1W + s2

)gµν

+ie2

2s2gµν

+ie2

2s2

(ZH Z2

g Z−2W − 1

)gµν


−ie2

2cgµν

−ie2

2c

(ZH Zg Z−1

W − 1)

gµν

+ie2

2sgµν

+ie2

2s

(ZH Zg Z−1

W − 1)

gµν

∓ e2

2cgµν

∓ e2

2c

(ZH Zg Z−1

W − 1)

gµν

± e2

2sgµν

± e2

2s

(ZH Zg Z−1

W − 1)

gµν

+ie2

2s2c2(1 − 2s2)2gµν

+ie2

2s2c2

[ZH(c

4 Z2g Z−2

W + s4

− 2s2c2 Zg Z−1W )− (1 − 2s2)2

]gµν

+i2e2gµν

+ie2

2

[ZH

(1 + Zg Z−1

W

)2 − 4]gµν

578 Appendix C

+ie2

sc(1 − 2s2)gµν

+ie2

2sc

[ZH(1 + Zg Z−1

W )(c2 Zg Z−1W − s2)

− 2(1 − 2s2)]gµν

+ie2

2s2gµν

+ie2

2s2

(ZH Z2

g Z−2W − 1

)gµν

Gauge interactions of the ghost fields

−gscabc pµ

−gs(ZηG ZS Z−1G − 1)cabc pµ

±iepµ

±ie(ZηL Zg Z−1W − 1)pµ

±iec

spµ

±iec

s(ZηL Zg Z−1

W − 1)pµ

∓iepµ

∓ie(ZηL Zg Z−1W − 1)pµ


∓iec

spµ

∓iec

s(ZηL Zg Z−1

W − 1)pµ

±iepµ

±ie(ZηL Zg Z−1W − 1)pµ

±iec

spµ

±iec

s(ZηL Zg Z−1

W − 1)pµ

Interactions of ghosts with Higgs and Goldstone bosons

−ie

2sξ MZ

−ie

2sξ MZ

[ZH

(1 − δv

v

)Zg Z−1

W

(c2 ZηL Zg Z−1

W

+ s2 Z1/2ηY

Z1/2ηL

Z−1/2W Z1/2

B

)− 1

]−iξ MW

−ie

2ξ MW ZH

(1 − δv

v

)Zg Z−1

W

(ZηL Zg Z−1

W

− Z 1/2ηY

Z1/2ηL

Z−1/2W Z1/2

B

)+i

e

2s

−ie

2sξ MZ

[ZH

(1 − δv

v

)Zg Z−1

W

(c2 ZηL Zg Z−1

W

− s2 Z1/2ηY

Z1/2ηL

Z1/2W Z−1/2

B

)− (1 − 2s2)

]

580 Appendix C

+ieξ MW

+ie

2ξ MW

[ZH

(1 − δv

v

)Zg Z−1

W

(ZηL Zg Z−1

W

+ Z 1/2ηY

Z1/2ηL

Z1/2W Z−1/2

B

)− 2

]

+ie

2sξ MW

+ie

2sξ MW

[ZH ZηL Z2

g Z−2W

(1 − δv

v

)− 1

]

∓ e

2sξ MW

∓ e

2sξ MW

[ZH ZηL Z2

g Z−2W

(1 − δv

v

)− 1

]

ie

2scξ MZ

+ie

2scξ MZ

[ZH

(1 − δv

v

)(s4 ZηY + c4 ZηL Z2

g Z−2W

+ s2c2 Z1/2ηY

Z1/2ηL

Zg(Z−3/2W Z1/2

B + Z−1/2W Z−1/2

B ))− 1

]

+ie

2ξscMZ ZH

(1 − δv

v

)×[ZηY + ZηL Z2

g Z−2W

− Z 1/2ηY

Z1/2ηL

Zg(Z−3/2W Z1/2

B + Z−1/2W Z−1/2

B )]

+ i

2eξ MZ ZH

(1 − δv

v

)×[s2(Z1/2

ηYZ1/2

ηLZg Z−1/2

W Z−1/2B − ZηY)

+ c2 Zg Z−1W (ZηL Zg Z−1

W − Z 1/2ηY

Z1/2ηL

Z−1/2W Z1/2

B )]


+ i

2eξ MZ ZH

(1 − δv

v

)× [

s2(Z1/2ηY

Z1/2ηL

Zg Z−3/2W Z1/2

B − ZηY)

+ c2 Zg Z−1W (ZηL Zg Z−1

W − Z 1/2ηY

Z1/2ηL

Z1/2W Z−1/2

B )]

We end this appendix with a few more remarks. We have chosen our conventionsto classify physical objects as particles and antiparticles described by chargeconjugate fields (for example, the electron is usually called a particle and describedby the field ψ and the positron is called an antiparticle and described by thecharge conjugate field ψc). A spin 1

2 particle state with momentum k and spinprojection s is created by the operator b+(k, s) and the antiparticle state bybc+(k, s) = d+(k, s). Using for instance, the rules of canonical quantizationintroduced in Sections 1.4 and 2.7 it is easy to see that a general fermion–bosonvertex i�, under interchange of incoming and outgoing particles (or antiparticles)is transformed into iγ0�

+γ0. This follows by taking the matrix element of theinteraction lagrangian

hi jkφi ψ′j�ψ ′

k + h∗i jkφ+i ψkγ0�

+γ0ψ j

between the appropriate one-particle states built of creation and annihilationoperators. Changing in the vertex particles (antiparticles) into antiparticles (par-ticles) gives the same transformation for the vertex. The vertex remains, ofcourse, unchanged under both transformations performed simultaneously. We alsorecall once more that each incoming (outgoing) fermion with momentum k andspin projection s is described by the spinor u(k, s) (u(k, s)) and each incoming(outgoing) antifermion with momentum k and spin projection s by v(k, s) (v(k, s)).

In some theories, for example, in the Minimal Supersymmetric Standard Model(MSSM), some fields appear in the lagrangian simultaneously with their chargeconjugate partners. For instance, in the MSSM, defining the positively chargedchargino χ+ as a particle, we have the couplings

χ+�ud∗ + (χ+)c�du∗ + h.c.

where u and d are quark fields and u and d are scalar fields of the respectivesupersymmetric partners of the quarks (called squarks). Both couplings may besimultaneously relevant in calculations of some amplitudes (for example, Z →χ+χ− at the one-loop level), so that the redefinition χ+ = (χ−)c does noteliminate the charge conjugate field from the Wick contractions. If χ c appearsto be an internal line then, after Wick contractions, we encounter the propagator

582 Appendix C

〈0|Tχ cχ c|0〉 = C〈0|Tχχ |0〉TC−1. In momentum space, C ST(p)C−1 = 1/(−/p−m) = S(−p), where the field χ carries momentum p. When the field χ and χ c

describe external particles like in Z → χ+χ− it is more convenient to rewrite oneof the interaction terms, for example,

χ c�du∗ = dc�′χ uc∗

where �′ = C�TC−1. Then, the charged conjugate fermion appears as an internalline and we can apply the simple propagator rule. A similar trick is useful inderiving Feynman rules for Majorana fermions (Denner 1992).

Appendix D

One-loop Feynman integrals

In this appendix we collect some useful formulae for integrals appearing in thecomputations of one-loop Feynman diagrams and general expressions for Feynmandiagrams contributing to the gauge boson self-energies.

Two-point functions

The two basic integrals are†

µε

∫dnk

(4π)n

i

k2 − m2≡ 1

(4π)2a(m)

µε

∫dnk

(4π)n

i

[k2 − m21][(k + p)2 − m2

2]≡ 1

(4π)2b0(p2,m1,m2)

(D.1)

where

a(m) = m2

(−η − 1 + ln

m2

µ2

)(D.2)

and

b0(p2,m1,m2) = −η +∫ 1

0dx ln

x(x − 1)p2 + xm21 + (1 − x)m2

2

µ2. (D.3)

Here µ is the renormalization scale and η = 2/(4 − n) + ln(4π) − γE . The b0-function is symmetric in m1 and m2. Useful limiting cases for the b0-function are

† The factor −iε in all masses squared is understood.

583

584 Appendix D

listed below:

b0(0,m1,m2) = −η − 1 + m21

m21 − m2

2

lnm2

1

µ2+ m2

2

m22 − m2

1

lnm2

2

µ2

= −η − 1 + lnm2

1

µ2+ m2

2

m22 − m2

1

lnm2

2

m21

= 1

m21 − m2

2

(a(m1)− a(m2)) (D.4)

b0(0, 0,m) = −η − 1 + lnm2

µ2(D.5)

d

dp2b0(0,m1,m2) = −1

2

m21 + m2

2

(m21 − m2

2)2+ m2

1m22

(m21 − m2

2)3

lnm2

1

m22

(D.6)

d

dp2b0(m

2,m, λ) = − 1

m2− 1

2m2ln

λ2

m2+O(λ2) (D.7)

Any other integral appearing in the computation of two-point Feynman diagramscan be reduced to these. For example,

µε

∫dnk

(4π)n

ikµ

[k2 − m21][(k + p)2 − m2

2]

= 1

(4π)2

pµ

2p2

[a(m1)− a(m2)− (p2 + m2

1 − m22)b0(p2,m1,m2)

](D.8)

µε

∫dnk

(4π)n

ikµkν

[k2 − m21][(k + p)2 − m2

2]

= 1

(4π)2

[gµν A(p2,m1,m2)+ pµ pν

p2B(p2,m1,m2)

](D.9)

where

A(p2,m1,m2) = 1

12a(m1)+ 1

12a(m2)+ 1

6

(m2

1 + m22 −

p2

2

)b0(p2,m1,m2)

+ m21 − m2

2

12p2

[a(m1)− a(m2)− (m2

1 − m22)b0(p2,m1,m2)

]− 1

6

(m2

1 + m22 −

p2

3

)(D.10)

One-loop Feynman integrals 585

B(p2,m1,m2) = −1

3a(m1)+ 2

3a(m2)− 1

3m2

1b0(p2,m1,m2)

+ (p2 + m21 − m2

2)2

3p2b0(p2,m1,m2)

− m21 − m2

2

3p2[a(m1)− a(m2)] + 1

6

(m2

1 + m22 −

p2

3

)(D.11)

Using the derivative (D.7) one easily finds

A(0,m1,m2) = 1

12a(m1)+ 1

12a(m2)+ 1

6(m2

1 + m22) [−1 + b0(0,m1,m2)]

+ 1

24

[m2

1 + m22 −

2m21m2

2

(m21 − m2

2)ln

m21

m22

](D.12)

In the limit m1 → m2 the last square bracket vanishes.The imaginary part of any two-point amplitude can easily be calculated knowing

that

Im b0(p2,m1,m2) = −πλ1/2(p2,m21,m2

2)θ

[1 − (m1 + m2)

2

p2

](D.13)

where λ(p2,m21,m2

2) is the standard phase space factor

λ(p2,m21,m2

2) = 1 − 2(m21 + m2

2)

p2+ (m2

1 − m22)

2

p4(D.14)

Three- and four-point functions

Three- and four-point functions for general non-vanishing external momenta andall masses different are very complicated. A compact expression exists only forthe scalar three-point function and involves twelve Spence functions (’t Hooft& Veltman 1979). Here we restrict ourselves to the case of vanishing externalmomenta. In this case the two basic integrals are∫

d4k

(2π)4

i

[k2 − m21][k2 − m2

2][k2 − m23]≡ 1

(4π)2F(m1,m2,m3)

(D.15)∫d4k

(2π)4

i

[k2 − m21][k2 − m2

2][k2 − m23][k2 − m2

4]≡ 1

(4π)2H(m1,m2,m3,m4)

(D.16)

Other integrals can be reduced to these two. Note that these are finite functions.They can be calculated by using the decomposition of the denominators into simple

586 Appendix D

fractions and then by using a(m)- or b0(0,m1,m2)-functions. For (D.15) we find

F(m1,m2,m3) = 1

m21 − m2

2

[b0(0,m1,m3)− b0(0,m2,m3)] (D.17)

In this form, the limit m21 → m2

2 is easily calculable by noting that (D.17) becomesin this limit just the definition of the derivative of b0(0,m1,m3) with respect to m2

1.Using the explicit form (D.4) of the b0 function we find

F(m1,m2,m3)

= 1

m21 − m2

2

(m2

1

m21 − m2

3

lnm2

1

m23

− m22

m22 − m2

3

lnm2

2

m23

)(D.18)

= −m21m2

2 ln(m21/m2

2)+ m22m2

3 ln(m22/m2

3)+ m23m2

1 ln(m23/m2

1)

(m21 − m2

2)(m22 − m2

3)(m23 − m2

1)(D.19)

Knowing the function F(m1,m2,m3), the function H(m1,m2,m3,m4) can berepresented as

H(m1,m2,m3,m4) = F(m1,m3,m4)− F(m2,m3,m4)

m21 − m2

2

(D.20)

Again, the limit m21 → m2

2 can be found by applying the trick with the derivative.As an application we compute the integral appearing in (12.139). After performingthe algebra of the Dirac matrices we are left with∫

d4k

(2π)4

ik2

[k2 − m2i ][k2 − m2

j ][k2 − M2

W ]2

=∫

d4k

(2π)4

[i

[k2 − m2i ][k2 − m2

j ][k2 − M2

W ]

+ iM2W

[k2 − m2i ][k2 − m2

j ][k2 − M2

W ]2

]= 1

(4π)2

[F(mi ,m j , MW )+ M2

W

d

dM2W

F(mi ,m j , MW )

]= 1

(4π)2 M2W

A(xi , x j )

where A(xi , x j ) in which xi ≡ m2i /M2

W defined in (12.143) is obtained by using(D.18) and xi ≡ m2

i /M2W .

General expressions for the one-loop vector boson self-energies

The general self-energy of a vector boson shown in Fig. 12.5(a) is decomposedinto the transverse and longitudinal parts according to (12.67). We define here


Fig. D.1. Gauge boson sector contributions to �µνV1,V2

(p2).

Fig. D.2. Matter sector contributions to �µνV1,V2

(p2).

another decomposition

i�µν

V1,V2(p2) = igµν�T

V1,V2(p2)+ i

pµ pν

p2�L

V1,V2(p2) (D.21)

so that �LV1,V2

≡ �LV1,V2

− �TV1,V2

. This decomposition is more convenientfor a concise presentation of the results of the calculation. Typical one-loopcontributions to �

µν

V1,V2(p2) are shown in Figs. D.1 and D.2. Evaluating diagrams

588 Appendix D

shown in Fig. D.1 we find

�T(a)(p2) = − g1g2

(4π)2

[10A(p2, M1, M2)+ (4p2 + M2

1 + M22 )b0(p2, M1, M2)

+ a(M1)+ a(M2)+ 2M21 + 2M2

2 − 23 p2

](D.22)

�L(a)(p2) = − g1g2

(4π)2

[10B(p2, M1, M2)− 2p2b0(p2, M1, M2)

− 5(a(M2)− a(M1)− (p2 − M2

1 + M22 )b0(p2, M1, M2)

)+ 23 p2

](D.23)

�T(b)(p2) = g1g2

(4π)2A(p2, M1, M2) (D.24)

�L(b)(p2) = g1g2

(4π)2

[B(p2, M1, M2)+ 1

2

(a(M2)− a(M1)

− (p2 − M21 + M2

2 )b0(p2, M1, M2))]

(D.25)

�T(c)(p2) = g2

(4π)2

[6a(M)+ 4M2

](D.26)

�L(c)(p2) = 0 (D.27)

For the diagrams shown in Fig. D.2 we get (the coupling of fermions to thevector boson is iγ κ(cV − cAγ

5))

�T(a)(p2) = 4

(4π)2(c2

V + c2A)[2A(p2,m1,m2)− 1

2 a(m1)

− 12 a(m2)+ 1

2(p2 − m21 − m2

2)b0(p2,m1,m2)]

+ 4

(4π)2(c2

V − c2A)m1m2b0(p2,m1,m2) (D.28)

�L(a)(p2) = 4

(4π)2(c2

V + c2A)[2B(p2,m1,m2)+ 1

2 a(m2)

− 12 a(m1)− (p2 − m2

1 + m22)b0(p2,m1,m2)

](D.29)

�T(b)(p2) = − 4

(4π)2g1g2 A(p2, M1, M2) (D.30)

�L(b)(p2) = − g1g2

(4π)2

[4B(p2, M1, M2)+ b0(p2, M1, M2)+ 2

(a(M2)

− a(M1)− (p2 − M21 + M2

2 )b0(p2, M1, M2))]

(D.31)

�T(c)(p2) = 1

(4π)2g2a(M) (D.32)

�L(c)(p2) = 0 (D.33)


�T(d)(p2) = 1

(4π)2λ1λ2b0(p2, M1, M2) (D.34)

�L(d)(p2) = 0 (D.35)

The functions a(m), b0(q2,m1,m2) and A(q2,m1,m2) are defined in (D.9).In the Standard Model, using the Feynman rules and taking proper account of

the combinatoric factors we find for �TW W (p2) (in units e2/(4π)2s2)

12

3∑k=1

[4A(p2,mek , 0)− a(mek )+ (p2 − m2

ek)b0(p2,mek , 0)

]+ 3

2

3∑k,l=1

|V klCKM|2

[4A(p2,muk ,mdl )− a(muk )− a(mdl )

+ (p2 − m2uk− m2

dl)b0(p2,muk ,mdl )

]− A(p2, MW , MH )− A(p2, MW , MH )+ 1

2 a(MW )+ 14 a(MZ )+ 1

4 a(MH )

+ s4 M2Z b0(p2, MW , MZ )+ s2 M2

W b0(p2, MW , 0)+ M2W b0(p2, MW , MH )

− s2[8A(p2, MW , 0)+ (4p2 + M2

W )b0(p2, MW , 0)+ 2a(MW )+ 23 p2

]− c2

[8A(p2, MW , MZ )+ (4p2 + M2

W + M2Z )b0(p2, MW , MZ )

− 2a(MW )− 2a(MZ )+ 23 p2

]For �T

Z Z (p2) (in units e2/(4π)2s2c2) we get:

34

[A(p2, 0, 0)+ p2b0(p2, 0, 0)

]+ 1

2

∑f

Nc f a f+[2A(p2,m f ,m f )− a(m f )+ ( 1

2 p2 − m2f )b0(p2,m f ,m f )

]+ 1

2

∑f

Nc f a f−m2

f b0(p2,m f ,m f )

− (c2 − s2)[A(p2, MW , MW )− 1

2 a(MW )]

− A(p2, MZ , MH )− 14 a(MZ )− 1

4 a(MH )

+ 2s4 M2W b0(p2, MW , MW )+ M2

Z b0(p2, MZ , MH )

− c4[8A(p2, MW , MW )+ (4p2 + 2M2

W )b0(p2, MW , MW )− 4a(MW )− 23 p2

]where the sum in the second line extends to all fermions except neutrinos, a f

+ =1− 4|Q f |s2 + 8|Q f |2s4, a f

− = −4|Q f |s2 + 8|Q f |2s4 and Q f , Nc f are the electriccharge and colour factor (1 for leptons, 3 for quarks) of the particle, respectively.

590 Appendix D

The expression for �Tγ γ (p2) in units e2/(4π)2 reads

2∑

f

Nc f Q2f

[4A(p2,m f ,m f )− 2a(m f )+ p2b0(p2,m f ,m f )

]− 4A(p2, MW , MW )− 2a(MW )+ 2M2

W b0(p2, MW , MW )

− [8A(p2, MW , MW )+ (4p2 + 2M2

W )b0(p2, MW , MW )− 4a(MW )− 23 p2

]Finally, �T

γ Z (p2) in units e2/(4π)2sc is given by

2∑

f

Nc f |Q f |(1 − 4|Q f |s2)[2A(p2,m f ,m f )− a(m f )+ p2b0(p2,m f ,m f )

]− (c2 − s2)

[2A(p2, MW , MW )− a(MW )

]− 2s2 M2W b0(p2, MW , MW )

− c2[8A(p2, MW , MW )+ (4p2 + 2M2

W )b0(p2, MW , MW )− 4a(MW )− 23 p2

]

Appendix E

Elements of group theory

For a concise introduction to group theory and its application in particle physicssee, for instance, Werle (1966). Here we recall only those definitions and propertiesmost useful in reading this book, with no attempt for a complete and self-consistentpresentation.

Definitions

Let H be a proper (i.e. different from the group G itself and from the unit elementalone) subgroup of G, and g an arbitrary element of G. The sets gH and Hg withfixed g ∈ G, are called left and right cosets of H . Two left cosets gi H and gk H(or two right cosets Hgi , and Hgk) of the same subgroup H contain exactly thesame elements of G or have no common elements at all. Taking all different left(or right) cosets of H we can decompose the group G into the sum

G = H + g1 H + · · · + gν−1 H (E.1)

where g1 ∈ G, g1 /∈ H , g2 ∈ G, g2 /∈ H , g2 /∈ g1 H etc. The number ν is calledthe index of H in G. We can define equivalence classes of elements of G; twoelements g and g′ are considered equivalent if they belong to the same coset withrespect to H . The set of all cosets is a manifold denoted by G/H .

Two elements g, h ∈ G are said to be mutually conjugate if there exists such anelement a ∈ G that

h = aga−1

One can divide all the group elements into separate classes of conjugate elements.If a set H with elements h is a subgroup of G then the set H ′ = gHg−1 is

another subgroup of G which is called conjugate to H . If for arbitrary g ∈ Gall the conjugate subgroups gHg−1 are identical (i.e. contain the same elements)gHg−1 = H , we call H a normal subgroup. A normal subgroup H of G consists of

591

592 Appendix E

whole undivided classes of G. Groups which contain no proper normal subgroupsare said to be simple. Groups which contain no proper normal abelian subgroupsare said to be semi-simple.

Consider the products of an element of the coset gH by an element of the cosetf H , where H is a normal subgroup of G. All such products belong to the cosetg f H . Hence we can define a new type of multiplication of cosets

(gH)( f H) = (g f H) (E.2)

The set of all different cosets of a normal subgroup H is a group with respect tothis multiplication law. This so-called quotient group F is not a subgroup of G asits elements are whole sets (cosets). We write symbolically F = G/H .

A function ϕ is said to be defined on a group G if to each element g ∈ G aunique complex number ϕ(g) is assigned. In the case of a continuous k-parametergroup G(k) a function ϕ(g) can be regarded as an ordinary function ϕ(α1, . . . , αk)

of k real variables that are necessary to specify the group elements. A group iscalled compact if the group manifold M is compact, i.e. if every infinite subset ofM contains a sequence converging to a limit in M , or equivalently if any functionϕ(αi ) which is continuous in all points of M is also bounded. For compact groupswe can define the average of a group function by the expression

Av[ϕ] = (1/V )

∫ϕ(α)ρ(α) dα (E.3)

where

V =∫

ρ(α) dα (E.4)

and ρ(α) is a properly chosen weight function such that

(i) Av[ϕ(g)] = c if ϕ(g) = const for all g ∈ G

(ii) Av[ϕ(g)] ≥ 0 if ϕ(g) ≥ 0

(iii) Av[ϕ(hg)] = Av[ϕ(gh)] = Av[ϕ(g)] for any fixed h ∈ G.

All representations of a semi-simple group are fully reducible. Unitary repre-sentations of any group are fully reducible. Every representation of a finite groupor of a compact Lie group is equivalent to a unitary representation. Lie groupswhich are not compact may have representations which are not equivalent to anyunitary representation. Any finite group has a finite number and any compactLie group has a countable infinite number of inequivalent, unitary, irreduciblerepresentations which are all finite-dimensional. Non-compact Lie groups haveuncountable numbers of inequivalent, unitary irreducible representations which areall infinite dimensional.

Elements of group theory 593

Transformation of operators

Consider the set of all possible linear operators O which transform the vectors |v〉of some Hilbert space H into vectors |w〉 of the same space |w〉 = O|v〉. Supposethat under a group of transformations G the vectors of H are transformed as follows

|wg〉 = U (g)|w〉, |vg〉 = U (g)|v〉 (E.5)

where U (g) = exp(iαa Qa) are unitary operators and Qa are generators of G. Theoperator Og which transforms |vg〉 into |wg〉 is given by

Og = U (g)OU−1(g) (E.6)

Let us define the linear transformation T (g) which transforms O → Og

Og = T (g)O = U (g)OU−1(g) (E.7)

The transformation T (g) = exp(−iαaT a) forms a unitary representation of G. Forinfinitesimal transformations we have

Og∼= (1 − iδαaT a)O = O + iδαa[Qa, O] (E.8)

Complex and real representations

Let Ti , i = 1, . . . , N form a representation of generators of a group G

[Ti , Tj ] = ici jk T k (E.9)

The structure constants ci jk are real if Ti are hermitean. Taking the complex conju-gate of these commutation relations we see that (−T ∗

i ) also form a representationof the group

[(−T ∗i ), (−T ∗

j )] = ici jk(−T k∗) (E.10)

If Ti and (−T ∗i ) are equivalent, i.e.

−T ∗i = U TiU

† (E.11)

where U is a unitary transformation, the representation is said to be real (the termreal is motivated by an alternative convention in which iTi is the representationmatrix). For instance for 2 and 2∗ of SU (2) we have

−T ∗i = −T T

i = − 12σ

Ti (E.12)

and

iσ2(− 12σ

Ti )(iσ2)

† = 12σi (E.13)

But this is not true for SU (N ), N > 2.

594 Appendix E

Take a set of n fields �a(x) which transform according to n × n-dimensionalrepresentation matrices Ti :

[Qi ,�a] = −(T i )ab�b (E.14)

Then the fields �†a transform according to the complex conjugate representation

(take the hermitean conjugate of (E.14)):

[Qi ,�†a(x)] = +T i∗

ab�†b(x) = −(−T i∗

ab )�†b(x) (E.15)

Let all the left-handed fermion fields �L and (�C)L (C means charge conjuga-tion, see Appendix A) in a theory form the representation fL under some symmetrygroup. From (A.110)

�R = C(�C)LT and (�C)R = (�L)

C = C�LT (E.16)

so �R and (�C)R transform under the complex conjugate representation f ∗L . If fL

is real then the left-handed and the right-handed fermions transform according tothe same representation of the symmetry group. The theory is said to be vector-like.If fL is complex ( fR = f ∗L �= fL) the theory is said to be chiral.

QCD is vector-like because

fL = 3 + 3∗

(uL)(uC)L(E.17)

while

fR = f ∗L = 3∗ + 3 = fL (E.18)

The standard model SU (3)× SU (2)×U (1) is chiral. For one family we have:

fL = (3, 2, 16)+ (3∗, 1,− 2

3)+ (3∗, 1, 13)+ (1, 2,− 1

2)+ (1, 1, 1)(ud

)L

(uC)L (dC)L(

ν

e−

)L

(e+)L (E.19)

and

fR = f ∗L = (3∗, 2,− 16)+ (3, 1, 2

3)+ (3, 1,− 13)+ (1, 2, 1

2) + (1, 1,−1) �= fL(dC

−uC

)R

uR dR(

e+

−νC

)R

e−R

(E.20)(We have used the equivalence 2∗ = 2 and the transformation rule (E.13)).

Traces

Let

[T a, T b] = icabcT c, cabc ≡ cabc (E.21)


Then

T2 =∑a,k

T aik T a

k j = δi j CR (E.22)

where R means representation R.

Tr[T aT b] = TRδab (E.23)

TRd(G) = CRd(R) (E.24)

where d(G) is the dimension of the group and d(R) is the dimension of therepresentation R.

For SU (N ):

d(R) = N for the fundamental representation (F)d(R) = N 2 − 1 = d(G) for the adjoint representation (A)

TF = 12 , CF = N 2 − 1

2NTA = CA = N∑

i,k caikcbik = δabCR = Nδab because in the adjoint representation (T a)ik =−icaik .

For SU (3) in the fundamental representation T a = 12λ

a with

λi =(σ i 00 0

), i = 1, 2, 3;

σ 1 =(

0 11 0

), σ 2 =

(0 −ii 0

), σ 3 =

(1 00 −1

)

λ4 = 1

01

λ5 = −i

0i

λ6 =(

0 00 σ 1

)

λ7 =(

0 00 σ 2

)λ8 = 1√

3

11

−2

(E.25)

{T a, T b} = 13δ

ab1l + dabcT c, dabc = dabc (E.26)

is totally symmetric. One has

1 = c123 = 2c147 = 2c246 = 2c257 = 2c345 = −2c156

= −2c367 = 2c458/√

3 = 2c678/√

3

596 Appendix E

1/√

3 = d118 = d228 = d338 = −d888

−1/2√

3 = d448 = d558 = d668 = d778

12 = d146 = d157 = d247 = d256 = d344 = d355 = −d366 = −d377

T aT b = 12(ic

abn + dabn)T n + 16δ

ab, Tr[T aT b] = 12δ

ab (E.27)

T aT bT c = 12(d

abn + icabn)T nT c + 16δ

abT c,

Tr[T aT bT c] = 14(d

abc + icabc) (E.28)

Tr[T aT bT cT d] = 18(d

abn + icabn)(dcdn + iccdn)+ 112δ

abδcd (E.29)

Tr[T aT bT aT b] = 18(d

abndabn − cabncabn)+ 812 = 1

8(d2 − f 2)+ 8

12 ,

d2 = 403 , f 2 = 24 (E.30)

σ -model

We derive the transformation properties of the π and σ fields in the σ -model. Weassume the nucleon doublets NR and NL transforming under the SUR(2)× SUL(2)as ( 1

2 , 0) and (0, 12) respectively. Therefore

under SUR(2)

NR → NR − iδαααR · ( 12σσσ)NR

NL → NL(E.31)

under SUL(2)

NL → NL − iδαααL · ( 12σσσ)NL

NR → NR(E.32)

(remember that σ aR NR = σ a⊗PR NR = σ a⊗(PR+PL)NR = σ a NR). For invariance

of the lagrangian (9.27) under each of these groups of transformations one needsthat the change of σ ′ + iπππ · σσσ compensates the change of the fermion fields:

σ ′ + iπππ · σσσ → σ ′ + iπππ · σσσ − iδ(σ ′ + iπππ · σσσ) (E.33)

where for SUL(2)

−iδ(σ ′ + iπππ · σσσ) = −iδαααL(12σσσ)(σ ′ + iπππ · σσσ)

= (−i 12σσσ)[σ ′δαααL + (πππ × δαααL)] + 1

2πππ · δαααL (E.34)

(σσσ · aσσσ · b = a · b + iσσσ · a × b)

and for SUR(2)

−iδ(σ ′ + iπππ · σσσ) = (σ ′ + iπππ · σσσ)(i 12σσσ)δαααR

= (−i 12σσσ)[−σ ′δαααR + (πππ × δαααR)] − 1

2πππ · δαααR (E.35)


Therefore we get the following transformation of fields:

SUL(2)

σ ′ → σ ′ + 12πππ · δαααL

πππ → πππ − 12πππ × δαααL − 1

2σ′δαααL

}(E.36)

SUR(2)

σ ′ → σ ′ − 12πππ · δαααR

πππ → πππ − 12πππ × δαααR + 1

2σ′δαααR

}(E.37)

The above results correspond to the commutation relations (9.30) (as can beimmediately seen using (9.32)). By taking appropriate linear combinations wealso get the relations (9.28).

Sometimes it is convenient to work in the real four-dimensional representation� ≡ (π1, π2, π3, σ ′)T. It is easy to check that

[Qa,�b] = −(T a)bi�i

where

T 1 =

−ii

T 2 =

i

−i

T 3 =

−i

i

(E.38)

5T 1 =

i

−i

5T 2 =

i

−i

5T 3 =

i−i

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Index

action, 12Euclidean, 83, 86quadratic in fields, 87

additive renormalization, 152Adler zero, 502Adler–Bardeen non-renormalization theorem, 468,

480Altarelli–Parisi equations, 312Altarelli–Parisi function, 306analytic continuation from Euclidean to

Minkowski space, 82, 86, 486angular momentum tensor, 19annihilation, electron–positron, 4, 249anomalies, 457

abelian, 480and the path integral, 478anomaly-free models, 478axial, 466cancellation in SU (2)×U (1), 475chiral, 468in Euclidean space, 486non-abelian and gauge invariance, 481, 491trace anomaly, 241

anomalous breaking of scale invariance, 238anomalous dimension matrix, 258, 267anomalous dimensions, 241, 245, 260anomalous divergence of the axial-vector current, 466anomalous magnetic moment of the electron, 193anti-BRS symmetry, 145antiunitary operator, 55Appelquist–Carazzone decoupling theorem, 174, 229asymptotic freedom, 5, 220, 274

and meson self-interaction, 228axial gauge, 132

background field, 122and anomalies, 478method in gauge theories, 279

bare quantities, 148Becchi–Rouet–Stora symmetry, 143Bianchi identity, 70Bjorken scaling, 254, 297

Cabibbo angle, 6, 400canonical dimensions, 159, 231canonical quantization, 35

and Feynman rules, 119and symmetries, 39, 42, 44annihilation operators, 37chiral fields, 44creation operators, 37Dirac fields, 43Majorana fields, 45normal ordering, 37scalar fields, 36Weyl fields, 41

canonical scaling, 215, 240causality, 38charge conjugation, 56, 552

eigenstates of, 61chiral anomaly, 468

for the axial U (1) current in QCD, 473chiral perturbation theory, 353chiral symmetry, 317

spontaneous breaking in strong interactions, 324,337

chiral theory, 25, 594classical currents, 15classical field equations, 20, 35, 107

solutions, 107, 546classical fields, 11, 20, 29colour, 3compact groups, 592complex representations, 593composite operators, 256

renormalization, 256, 445conformal algebra, 269

current, 236transformations, 235

connected Green’s function, 105connected proper vertex function, 105conserved charges and constants of motion, 15consistent and covariant anomaly, 384convention for the logarithm, 561coset, 498, 592Coulomb gauge, 132

605

606 Index

Coulomb scattering, 200counterterms, 152

and Feynman rules, 164and the renormalization conditions, 152in the Standard Model, 409, 563

covariant derivative, 30, 500, 519covariant gauge, 130

and ghosts, 140CP transformation, 25CP violation, 64, 401CPT theorem, 62

Dashen’s relation, 356Dashen’s theorems, 356deep inelastic hadron leptoproduction, 250

Bjorken scaling, 254 r , 429 ρ, 416dilatation current, 233dimensional regularization, 166, 184dimensional transmutation, 242, 376Dirac algebra, 558Dirac equation, 27

solutions, 548Dirac mass term, 24Dirac spinors, 26, 544discrete symmetries, 48dispersion calculations in QED, 196divergent diagrams

classification, 159necessary counterterms, 161

double logarithms, 194dual field strength tensor, 69dynamical breaking of gauge symmetries, 379

effective action, 107and spontaneous symmetry breaking, 109

effective coupling constant, 215, 222in QED, 223, 438

effective field theories, 173, 262effective four-fermion interactions, 402, 435, 447

muon decay, 403, 422QED corrections to muon decay, 404

effective lagrangian for FCNC transitions, 447effective lagrangians, 495

and abelian anomaly, 505and non-abelian anomaly, 506

effective low energy electroweak theory, 435effective potential, 111, 122

for λ�4, 112, 122electron form-factor, 206electron mass in M S scheme, 189electron self-energy, 187electroweak theory, 5, 389

covariant gauges, 396currents, 394CP violation, 401 ρ, 416fermion self-energies, 418gauge boson propagators, 411lagrangian, 392

physical gauge bosons, 394renormalization, 407running α(µ), 419, 440tree level, 401

electroweak unification, 7energy-momentum tensor, 18, 234εK , 66, 454equations of motion, 13

for scalar fields, 20for Dirac fields, 27, 548for gauge fields, 31, 35for Weyl fields, 24, 546

Euclidean action, 83, 86Euclidean Dirac operator and its eigenvalue problem,

487Euclidean Green’s functions, 82Euclidean path integral and anomalies, 486evanescent operator, 424

factorization theorem in QCD, 291, 304Faddeev–Popov determinant, 124, 133FCNC transitions, 441Fermi constant, 6, 402, 428Fermi interaction, 6, 402, 422fermion generations, 389fermion masses and mixing, 398fermion propagator, 102Feynman gauge, 132, 373Feynman integrals, 559, 583Feynman propagator, 39, 102

for gauge bosons, 411Feynman rules and canonical quantization, 119Feynman rules for

λ�4, 555QCD, 133, 557QED, 556Standard Model, 563

Fierz rearrangement, 424, 451fixed points, 219functional derivative, 76

γ matrices, 27, 545chiral representation, 545Dirac representation, 545

gauge dependence of the β-function, 227gauge invariance, 1

and the path integral, 124gauge transformations

abelian, 30non-abelian, 31

Gaussian integration, 88for Grassmann variables, 101integrals, 560

generating functionals, 90, 105ghosts, 140GIM mechanism, 443global symmetries and concervation laws, 13Goldberger–Treiman relation, 323Goldstone bosons, 327

classification, 380in QCD, 337

Index 607

Goldstone’s theorem, 321, 330, 337graded Lie algebra, 511Grassmann variables, 100Green’s functions, 79, 81, 83

and the scattering operator, 113as path integrals, 83boundary conditions for, 80derivatives of, 85equation of motion for, 121Euclidean, 82, 86for harmonic oscillator, 121for rescaled momenta, 214

GUTs, 268

Hamilton’s principle, 12harmonic oscillator, 78hierarchical symmetry breaking, 378Higgs mechanism, 369, 390, 392

and gauge boson propagator, 371homotopy classes, 282

imaginary time formalism and vacuum-to-vacuumtransitions, 76

index theorem for the Dirac operator, 490instantons, 286, 289IR problem in

QCD, 314QED, 200

Jacobi identityfor covariant derivatives, 70in graded Lie algebras, 512

K 0–K 0 system, 54

Landau–Cutkovsky rule, 141, 208leading logarithm approximation, 313Lehman–Symanzik–Zimmermann reduction theorem,

118light-cone

and deep inelastic hadron leptoproduction, 249expansion, 247gauge, 133parametrization of the four-momenta, 291

loop expansion, 99, 112Lorentz gauge, 129Lorentz transformations, 19, 23, 539

magnetic moment of the electron, 193Majorana mass term, 24Majorana spinor, 27minimal subtraction scheme, 156

in λ�4, 170in QED, 186in electroweak theory, 408

M S renormalization scheme, 186and fine structure constant, 186, 226, 438

neutral current transitions, 6, 405neutrino masses, 393, 399Noether current, 18, 70

Noether’s theorem, 18non-abelian gauge symmetry, 31non-covariant gauge, 132non-linear realization of the symmetry group, 495

σ -model, 495non-renormalization theorem in supersymmetrictheories, 531normal subgroup, 592normalization of one-particle state and ‘physical’

fields, 117

one-loop Feynman integrals, 583two-point functions, 583three- and four-point functions, 585

one-loop vector boson self-energies, 586one-particle states, 37, 42, 44, 117one-particle-irreducible Green’s function, 105operator product expansion (OPE), 243

and effective field theories, 262overlapping divergences, 163

parity, 550parton model, 290path integrals

and anomalies, 478and decoupling, 263for fermions, 100in Euclidean space, 486in quantum field theory, 83in quantum mechanics, 71

Pauli–Lubansky vector, 19PCAC, 323, 327perturbation theory

and generating functional, 90in momentum space, 97introduction, 90

photon propagator, 185, 186‘physical’ field, 117, 148π+–π0 mass difference, 362π0 decay, 471Poisson bracket, 15proton decay, 1

QCD, 3, 272asymptotic freedom, 274BRS transformaton, 273group theory factors, 275, 595lagrangian, 272perturbative, 290Slavnov–Taylor identities, 275

QCD corrections to CP violation, 445QED, 177

as effective theory, 438IR problem, 200lagrangian, 177massless, 194radiative corrections, 184Ward–Takahashi identities, 179

608 Index

quarkelectroweak interactions, 389masses, 353model, 3structure functions, 296

Rξ gauge, 372, 397radiative corrections in QED, 184renormalizability, 160renormalization, 148

arbitrariness, 153constants, 150in electroweak theory, 407in λ�4, 164of composite operators, 256of currents, 351scheme dependence, 224

renormalization group, 209and operator product expansion, 256in electroweak theory, 435, 438, 449in QED, 215

renormalization group equation, 209,for Wilson coefficients, 259

renormalization group functions β, γ , γm , 216renormalization of local composite operators, 256,

445RGE for, 258

renormalized Green’s functions and the S-matrix, 155renormalized lagrangian, 152renormalized perturbation theory, 155, 164renormalized quantities, 154RGE for Wilson coefficients, 259, 445

scalar field theory, 20scale dimensions, 244scale invariance, 230

anomalous breaking, 238soft breaking, 242

scattering operator, 116Schwinger terms, 348, 246seagull terms, 348self-energy

electron, 187fermion, 418photon, 185W and Z bosons, 411

semisimple group, 592short distance expansion, 243σ -model, 324, 596,

non-linear, 495simple group, 592Slavnov identities, see Ward–Takahashi identitiessoft symmetry breaking, 242, 344space reflection, 48, 550space-time transformations and conservation laws, 16Spence functions, 562spinor fields, 22

chiral, 26Dirac, 26Majorana, 27two-component (Weyl), 23

spinors, 539spontaneous chiral symmetry breaking

in QCD, 338in the σ -model, 325, 336

spontaneous symmetry breaking, 21and effective action, 109by radiative corrections, 373in SU (2)×U (1), 336, 390in supersymmetric theories, 513, 524patterns, 333

Standard Model, 389structure functions in deep inelastic scattering, 293Sudakov form-factor, 208Sudakov technique, 311sum rules for current matrix elements, 363superdeterminant, 102, 123superficial degree of divergence, 158

in n dimensions, 176superfields, 519supergraphs, 531superpropagators, 535superrenormalizability, 160superspace, 515supersymmetric covariant derivative, 519supersymmetric lagrangian, 521supersymmetry, 17supersymmetry algebra, 511

generators, 518supersymmetry breaking, 513, 524supertrace, 122supertranslation, 517symmetry transformations on quantum fields, 39, 42,

44, 45, 50, 55, 60

θ -vacuum, 286’t Hooft–Feynman gauge, 373, 397Thomson scattering, 420time reversal, 53, 551top quark, 7topological charge, 287topological vacua, 282T -product ambiguitytriangle diagram

calculation, 459renormalization constraints, 464

twist, 248two-component notation, 23, 539two-dimensional representations of SL(2,C), 539

UA(1) problem, 289, 473unitarity and ghosts, 140unitary gauge, 370

vacuum alignment, 379vacuum degeneracy, 325vacuum polarization, 184, 411vacuum structure in non-abelian gauge theories, 282vacuum-to-vacuum transitions and the imaginary time

formalism, 76vector-like theory, 25, 594

Index 609

vertex correctionsin electroweak theory, 427, 433in QED, 191, 195, 198

vielbein, 500

W boson mass, 405, 429Ward–Takahashi identities

and short-distance singularities of the anomalous,463

general derivation from path integrals, 344in QCD (Slavnov–Taylor identities), 277in QED, 179operator product, 348

weak hypercharge, 389Weinberg angle, 9, 402Wess–Zumino consistency conditions, 484Wess–Zumino model, 28, 521

Wess–Zumino term, 506Weyl equations, solutions, 546Weyl spinors, 541Wick’s theorem, 92, 114Wilson coefficients, 244, 264

and moments of the structure function, 254for FCNC, 447renormalization, 449

winding number, 283

Yukawa couplings, 29, 393

Z1 = Z2in electroweak theory, 410in QED, 178, 183

Z0 couplings, 406Z0 partial widths, 430

Erratum for the Book:

S. Pokorski, Gauge Field Theories

(2nd Edition)

Cambridge University Press

Last update: March 25, 2008

Place: Instead of: Should be:

p.8,line 2 in eq. (0.9): =

∑f ΨfQγµΨf =

∑f ΨfQγµΨf

p. 17,line 2 in eq. (1.35): = Lδxµ + ∂L

∂(∂µΦ)δ0Φ + δΛµ + ... = ∂µ[Lδxµ + ∂L

∂(∂µΦ)δ0Φ + δΛµ] + ...]

p. 69,last line: +2

3AσAµAν −2

3AσAµAν

p. 85,eq. (2.56a) and next eq. −iδ(x− y) −ihδ(x− y)

p. 109,lines below eq. (2.153): It can be proved, for example, It can be proved that ...

by induction, that ...

line 1 below eq. (2.154): The e�ective action ... An easy way to prove eq. (2.154) isto consider the relation between thefunctionals W [J ] and W ′[J ] for the�elds Φ and Φ, respectively, andsimilarly Z[J ] and Z ′[J ] †. Thee�ective action ...† I thank Yukinao Akamatsu, University

of Tokyo, for pointing it to me.

p. 110,2 lines above eq. (2.156): Also, we can prove, for Similarly to eq. (2.154) one can

example, by induction, that ... prove that ...

p. 111,line 3 from the bottom: (2.155) and (2.156) (2.163) and (2.164)

p. 113,eq. (2.168) hI(t) = ... HI(t) = ...

p. 133,eq. (3.38): gcαβγAµγ(x)∂µ)...(Dµ∂

µ)αβ... gcαβγ∂(x)µ Aµγ(x))...(∂µD

µ)αβ...

1


p. 135,line 2 in eq. (3.46): Aµγ(x)∂µ]... ∂µA

µγ(x)]...

line 2 in eq. (3.48): gcαβγAγµη∗α(∂µηβ)] gcαβγ(A

γµη∗α(∂µηβ) + (∂µAγµ)η∗αηβ)]

p. 144,line 2 from the bottom: = (Mη)α = 1

α(Mη)α

p. 148,line 6 from the bottom: mess mass

p. 149,line 2 in eq. (4.3): (y0 − x0) Θ(y0 − x0)line 5 in eq. (4.3): i

k2−m2F+iε

ik2−m2

F+iε

p. 168,eq. (4.50): dx

∫[...] dx[...]

eq. (4.53): 316π2ε

1ε

316π2

1ε

p. 169,line 2 in eq. (4.55): = I ′n + 1

2(n− 2)In = −1

2I ′n − 1

2(n− 6)In

line 6 from the bottom: 1 < n < 5 1 < n < 6

p. 170,line 2 in eq. (4.59): λ

16π2 = 1ε

= λ16π2

1ε

p. 171,last eq.: + 1

ε2[...] +1

ε[...]

p. 177,eq. (5.1): −1

4FBµν + F µν

B + ... −14FBµνF

µνB + ...

p. 180,line 2 in eq. (5.12): α = δΓ

δΨα = − δΓ

δΨ

p. 181,eq. above (5.18): = − δ

δα(z)δ

δα(z)= − δ

δα(z)δ

δα(x)

line 2 in eq. (5.18): +iqe δ2

δα(x)δα(y)δ(y − x) +iqe δ2

δα(x)δα(y)δ(y − z)

p. 185,line 1 below eq. (5.26): ... reads ... reads (see p. 266 for the

discussion of the running α(q2))

p. 188,eq. (5.47): 1

(p+k)2m21

(p+k)2−m2

p. 189,line 1 in eq. (5.49): = 2α

π1ε

= −2απ

1ε

line 2 in eq. (5.49): (Z0 − 1)(1) = ... (Z2 − 1)(1) = ...

2

Place Instead of: Should be:

p.191,eq. (5.61): ... 1

k2 (Z1 − 1)(1)γµ ... 1k2 + (Z1 − 1)(1)γµ

p. 193,eq. (5.68): u(p′)γµu(p)− ... u(p′)γµu(p) + ...eq. (5.70): F2(q2)... F2(q2) = ...

p. 199,line 7 from the bottom: [Γ(1 + 1

2ε)Γ(1

2ε)/Γ(1 + 1

2ε) [Γ(1 + 1

2ε)Γ(1

2ε)/Γ(1 + ε)

−Γ2(1 + 12ε)/Γ(2 + ε)] −2Γ2(1 + 1

2ε)/Γ(2 + ε)]

p. 200,line below eq. (5.100): Ze2 Zenext formula: ... 1

2|p| ... = 12|p|

p. 201,line below eq. (5.102): = 2|p|2 cos θ = |p|2 cos θ

p. 202,

eq. (5.108): d3p′

(2π)3d3p′

2p′0(2π)3

p. 211,eq. (6.6): Z3

3Z−11 ln(Z2

3Z−11 )

p. 212,line 2 in eq. (6.11):

∫ 10 ...

∫ t0 ...

p. 213,eq. (6.16):

∫ 10 ...

∫ t0 ...

p. 214,line 2 below eq. (6.20): the 1PI Green's function Γ(n) by ... the 1PI Green's function Γ(n) (in

a complete discussion one shouldinclude 1P reducible Green'sfunctions as well; such an extensionis straightforward) by ...

p. 216,line 2 in eq. (6.28): γm(α, a) = limε→0 ... γm(α, a) = − limε→0 ...lines below eq. (6.28): mB = Z−1

m m, ...α = Z−13 aB mB = Zmm, ...a = Z−1

3 aB

p. 218,

eq. (6.38) for Z0: − λ2

2(16π2)ε− λ2

2(16π2)2ε

eq. (6.38) for Z3: − λ2

12(16π2)ε− λ2

12(16π2)2ε

last equation: b2 + ... b22ε

+ ...

p. 225,eq. (6.61) and above it: b0 ... b1 β0 ... β1

3


p. 245,line 2 in eq. (7.60): = − i

4π2 ... = + i4π2 ...

p. 291,line 2 in eq. (8.69): P = (zP + (p2

⊥ + p2⊥)... p = (zP + (p2

⊥ + p2)...

p. 319,line 3 in eq. (9.9): Qb

R QbR,L

p. 324,line in eq. (9.26): −1

4γ... −1

4λ...

p. 339,eq. (9.87): = v3σ

3 +v3σ3

p. 345,line 1 above eq. (10.13): de�ned as ... are de�ned as ...line 1 in eq. (10.13): +∂µΘa(x) + ... −∂µΘa(x) + ...

p. 346,eq. (10.15): +cabc... −cabc...

p. 362,line 1 above eq. (10.80) Tjµ(x)jµ(x) Tjµ(x)jµ(0)and in eq. (10.80):

p. 398,line 1 in eq. (12.34): Mab

LL(x, y)ηaL(y) MabLL(x, y)ηbL(y)

p. 400,line 4 from the top: MBAνcAL νcBL MABνcAL νcBLeq. (12.41): YνM

−1Y Tν Y T

ν M−1Yν

p. 411,line 2 from the bottom: and process-dependent and corrections, often called

corrections consisting ... process-dependent, consisting ...

p. 412,starting at line 5 the splitting into universal and the splitting into universal andunder Fig. 12.5: process-dependent parts is so-called process-dependent parts

gauge-dependent and to get is gauge-dependent: the vertexphysical ... and box corrections contain also

contributions which are independentof the external legs and cancelthe gauge dependence of thecorrections to the gauge bosonpropagators. To get physical ...

4


p. 412,line 13 under Fig. 12.5: of at least part of the of the external leg independent

process-dependent corrections ... vertex corrections ...

line 1 in the footnote‡: is cancelled by the vertex is cancelled by the externalcorrections. ... leg independent part of

vertex corrections. ...

p. 461,line 1 below eq. (13.9): pνενδαβp

αqβ pµενδαβpαqβ

p. 541,

eq. (A20): χα = χβεβα χα = χβε

αβ

χα = χβεβα χα = χβεαβ

p. 565,line 3 from the top: +is2c2M2

ZZH(1− δvv

)2(1− ZgZ−1W )2gµν erase this line

p. 567,propagator of η+: η+...η+ η±...η±

p. 579,vertex G±η∓ηγ: −iξMW erase this linevertex G±ηZη±: +i e

2s+i e

2sc(c2 − s2)ξMW

p. 581,line 19 from the bottom: ψ′jΓψ

′k ψjΓψk

p.583, 584,eqs. (D.1), (D.8), (D.9): dnk

(4π)ndnk

(2π)n

5

Stefan Pokorski-Gauge Field Theories and Errata

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